MGU Mathematics Series — Algebra II
Part IV: Exponential, Logarithmic, and Sequence Models
Sequences · Series · Sigma Notation · Compound Interest · Applications
Chapter 11 of 14
MGU Mathematics Series  ·  Algebra II  ·  Library Textbook Edition

Chapter 11:
Sequences
and Series

Ordered patterns, cumulative sums, arithmetic and geometric sequences, sigma notation, and models from finance, biology, and computing

■ Math Level: Algebra II ■

■ 14 Sections  •  Part IV of 5 ■

Follows Chapter 10: Logarithmic Functions  ·  Prepares students for Chapter 12: Conic Sections and beyond

Chapter 11 — Front Matter

Purpose and Overview

This chapter introduces sequences as ordered lists that follow patterns and series as the sums of those lists. Sequences appear wherever quantities change in a predictable, step-by-step way: the yearly balance of a savings account, the number of cells in a dividing culture, the distances a bouncing ball travels, the rows in a triangular arrangement of objects. In each case, there is a starting value, a rule for moving from one term to the next, and a structure that can be described with algebra.

Sequences are, in an important sense, functions. Instead of accepting any real number as input, a sequence accepts only counting numbers — 1, 2, 3, 4, and so on — as positions. The output is the value at each position. This makes sequences the discrete cousins of the continuous functions studied throughout Algebra II. A linear function describes a line; an arithmetic sequence describes the same kind of additive growth in steps. An exponential function describes continuous multiplicative growth; a geometric sequence describes the same growth in discrete jumps.

By the end of this chapter, students should be able to recognize and write sequences, use explicit and recursive formulas, compute sums of arithmetic and geometric series, work fluently with sigma notation, and apply these tools to compound interest, science, computing, and other fields. This chapter also prepares students for the deeper treatment of series and limits in Precalculus, Calculus, and Discrete Mathematics.

Opening Question

A student deposits $1,000 in a savings account that earns 5% annual interest, compounded once per year. After the first year, the account holds $1,050. After the second year, it holds $1,102.50. After the third year, it holds $1,157.63. Is this pattern linear? Is the increase the same every year? If the student wants to know how much money will be in the account after 20 years, is there a formula that gives the answer directly, without computing 20 separate steps? These questions lead directly to sequences and series.

14Sections
11Objectives
20Key Terms
15+Worked Examples

Learning Objectives

  1. 1Define a sequence and identify its terms using index notation correctly.
  2. 2Write explicit formulas and use them to find any term of a sequence directly.
  3. 3Write recursive formulas and use them to generate sequences step by step.
  4. 4Identify arithmetic sequences by their common difference and write their explicit formulas.
  5. 5Compute partial sums of arithmetic series.
  6. 6Identify geometric sequences by their common ratio and write their explicit formulas.
  7. 7Compute partial sums of finite geometric series.
  8. 8Determine when an infinite geometric series converges and find its sum.
  9. 9Read and write sigma notation for series.
  10. 10Apply sequences and series to compound interest, population growth, science, and computing.
  11. 11Recognize and avoid common mistakes involving sequences and series.
Chapter 11 — Contents

Sections at a Glance

Sections 11.1 – 11.4
Sequences: Ordered Patterns and Their Formulas
What a Sequence Is · Terms and Indexing · Explicit Formulas · Recursive Formulas
Section 11.1

What a Sequence Is

A sequence is an ordered list of numbers that follows a rule. The numbers in the list are called terms. What makes a sequence different from a simple set of numbers is that order matters — each term occupies a specific position in the list, and that position is part of what defines the sequence.

Here are four examples of sequences:

SequencePattern
$2, 5, 8, 11, 14, \ldots$Each term is 3 more than the one before it
$3, 6, 12, 24, 48, \ldots$Each term is twice the previous one
$1, 1, 2, 3, 5, 8, 13, \ldots$Each term is the sum of the two terms before it
$10, 7, 4, 1, -2, \ldots$Each term is 3 less than the one before it

The three dots at the end of a sequence — called an ellipsis — mean "and so on according to the same pattern." A finite sequence has a last term. An infinite sequence continues without end.

A sequence is a function whose domain is a set of positive integers. Instead of writing $f(x)$ for a function of a continuous variable, mathematicians write $a_n$ for the value of a sequence at position $n$. Here, $n$ is the index — an integer identifying the position of the term.

Sequence as a Function

A sequence $a_n$ is a function whose domain is the set of positive integers. The index $n$ is the input (position); the term $a_n$ is the output (value).

The subscript $a_n$ is read "a sub n" or "a subscript n." It refers to the term at position $n$. The subscript is not a power — it is a label for position.

Student Warning — Subscript vs. Exponent

The term $a_3$ is the third term of the sequence — it is not $a$ raised to the power 3. Context will always make this clear, but pay close attention to where the number is written: a subscript labels a position, while a superscript (exponent) indicates a power.

Section 11.2

Terms and Indexing

Every term of a sequence has a position, identified by its index. By convention, sequences in this textbook begin with index $n = 1$ unless otherwise stated. Some sequences in advanced mathematics and computer science begin with index $n = 0$; for most of Algebra II, assume sequences start at $n = 1$ unless explicitly told otherwise.

Position (index)$n = 1$$n = 2$$n = 3$$n = 4$$n = 5$
Sequence $4, 7, 10, 13, 16, \ldots$$a_1 = 4$$a_2 = 7$$a_3 = 10$$a_4 = 13$$a_5 = 16$
Example 11.2.1
Identifying terms from a list
  • ProblemA sequence begins: $3, 9, 27, 81, 243, \ldots$ Identify $a_1$, $a_3$, and $a_5$. Describe the pattern in words.
  • ThinkRead the terms in order. The first term corresponds to index 1, the second to index 2, and so on.
  • Solve$a_1 = 3, \quad a_3 = 27, \quad a_5 = 243$
  • Check$3 \times 3 = 9$, $9 \times 3 = 27$, $27 \times 3 = 81$, $81 \times 3 = 243$. ✓
Answer: $a_1 = 3$, $a_3 = 27$, $a_5 = 243$. Each term is three times the previous term.
Practice 11.2Terms and Indexing
  1. What is the difference between the index of a term and the value of a term?
  2. If a sequence is a function, what is its domain? What is its range?
  3. A sequence begins with $a_1 = 7$. Does this tell you what $a_2$ is? Explain.
  4. List the first five terms of $5, 10, 15, 20, 25, \ldots$, then identify $a_3$ and $a_5$.
  5. Write the terms $a_1, a_2, a_3, a_4$ for the sequence $100, 50, 25, 12.5, \ldots$
  6. Construct a table of values for $n = 1, 2, 3, 4, 5$ for the sequence $-3, -1, 1, 3, 5, \ldots$
Section 11.3

Explicit Formulas

An explicit formula gives the value of any term of a sequence directly, using only the index $n$. It does not require knowing previous terms. Given an explicit formula, you can jump immediately to the 50th term, the 100th term, or any other term simply by substituting the index. An explicit formula is written $a_n = \text{(expression involving } n\text{)}$.

Why Explicit Formulas Are Powerful

Without an explicit formula, finding the 200th term requires computing all 199 previous terms. With an explicit formula, substitute $n = 200$ and evaluate immediately. This is one of the great advantages of algebraic formulas over arithmetic iteration.

Example 11.3.1
Using an explicit formula
  • ProblemThe explicit formula for a sequence is $a_n = 3n + 1$. Find $a_1$, $a_5$, and $a_{10}$.
  • ThinkSubstitute each desired index into the formula and compute.
  • Solve $a_1 = 3(1) + 1 = 4$ $a_5 = 3(5) + 1 = 16$ $a_{10} = 3(10) + 1 = 31$
  • CheckThe sequence begins $4, 7, 10, 13, 16, \ldots$ Each term increases by 3. The 10th term should be $4 + 9(3) = 31$. ✓
Answer: $a_1 = 4$, $a_5 = 16$, $a_{10} = 31$.
Example 11.3.2
Writing an explicit formula from a table
  • ProblemThe table shows the first four terms. Write an explicit formula: $a_1 = 5$, $a_2 = 8$, $a_3 = 11$, $a_4 = 14$.
  • ThinkThe terms increase by 3 each time — a linear pattern. A linear explicit formula has the form $a_n = mn + b$.
  • SolveSince the terms increase by 3 per step, $m = 3$. When $n = 1$: $3(1) + b = 5$, so $b = 2$. $a_n = 3n + 2$
  • Check$a_3 = 3(3)+2 = 11$ ✓ and $a_4 = 3(4)+2 = 14$ ✓.
Answer: $a_n = 3n + 2$.
Example 11.3.3
An explicit formula involving powers
  • ProblemA sequence has the explicit formula $a_n = 2^n$. List the first five terms.
  • Solve $a_1 = 2^1 = 2, \quad a_2 = 4, \quad a_3 = 8, \quad a_4 = 16, \quad a_5 = 32$
Answer: $2, 4, 8, 16, 32, \ldots$
Practice 11.3Explicit Formulas
  1. Use $a_n = 4n - 1$ to find $a_1, a_4, a_7$, and $a_{20}$.
  2. Use $a_n = (-1)^n \cdot n$ to list the first six terms.
  3. Use $a_n = \frac{1}{n}$ to list the first five terms. What happens as $n$ increases?
  4. Write an explicit formula for $6, 11, 16, 21, 26, \ldots$
  5. Write an explicit formula for $100, 90, 80, 70, 60, \ldots$
  6. A theater has rows of seats. Row 1 has 12 seats; each additional row has 3 more. Write an explicit formula for row $n$. How many seats are in row 15?
Section 11.4

Recursive Formulas

A recursive formula defines each term of a sequence by referring to one or more previous terms. Instead of computing $a_n$ directly from the index alone, you must know the preceding term and apply the rule repeatedly. A recursive formula has two parts: a starting value such as $a_1 = 5$, and a recurrence rule such as $a_n = a_{n-1} + 3$ for $n \geq 2$. Together, these completely define the sequence.

Recursive formulas are natural in many real situations. When the value of something tomorrow depends on what it is today, a recursive formula models that relationship directly. The tradeoff is that finding a distant term requires computing every term in order.

Example 11.4.1
Generating a sequence from a recursive formula
  • Problem$a_1 = 2$ and $a_n = a_{n-1} + 5$ for $n \geq 2$. List the first five terms.
  • Solve $a_1 = 2, \quad a_2 = 7, \quad a_3 = 12, \quad a_4 = 17, \quad a_5 = 22$
Answer: $2, 7, 12, 17, 22, \ldots$
Example 11.4.2
The Fibonacci sequence — two-term recursion
  • Problem$a_1 = 1$, $a_2 = 1$, and $a_n = a_{n-1} + a_{n-2}$ for $n \geq 3$. List the first eight terms.
  • ThinkThis recursive rule requires two previous terms. Each new term is the sum of the two immediately before it.
  • Solve $a_1 = 1, \; a_2 = 1, \; a_3 = 2, \; a_4 = 3, \; a_5 = 5, \; a_6 = 8, \; a_7 = 13, \; a_8 = 21$
Answer: $1, 1, 2, 3, 5, 8, 13, 21, \ldots$
Fibonacci in the Real World

The Fibonacci sequence appears in the branching of plants, the spiral arrangement of seeds in sunflowers, the proportions of classical composition in art, and fundamental algorithms in computer science. Its recursive definition is remarkably simple; its applications are vast.

Practice 11.4Recursive Formulas
  1. Use $a_1 = 10$, $a_n = a_{n-1} - 4$ to list the first six terms.
  2. Use $a_1 = 1$, $a_n = 3 \cdot a_{n-1}$ to list the first five terms.
  3. Use $a_1 = 100$, $a_n = a_{n-1} \cdot \frac{1}{2}$ to list the first five terms.
  4. Write a recursive formula for $4, 7, 10, 13, 16, \ldots$
  5. Write a recursive formula for $5, 15, 45, 135, 405, \ldots$
  6. A bacteria culture starts with 200 cells and doubles each hour. Write a recursive formula and find the count after 5 hours.
Sections 11.5 – 11.6
Arithmetic Sequences and Series
Common Difference · Explicit Formula · Partial Sums · The Gauss Method
Section 11.5

Arithmetic Sequences

An arithmetic sequence is a sequence in which each term is obtained by adding a fixed number to the previous term. That fixed number is called the common difference, denoted $d$. The common difference can be positive (increasing), negative (decreasing), or zero (constant).

Identifying an Arithmetic Sequence

Subtract each term from the term that follows it. If the difference is constant, the sequence is arithmetic.

$d = a_2 - a_1 = a_3 - a_2 = a_4 - a_3 = \cdots$

The Explicit Formula for an Arithmetic Sequence

Arithmetic Sequence — Explicit Formula
$$a_n = a_1 + (n-1)d$$

To reach term $n$, start at $a_1$ and add the common difference exactly $(n-1)$ times. This is the discrete counterpart of a linear function: $d$ plays the role of slope.

Example 11.5.1
Finding terms of an arithmetic sequence
  • ProblemAn arithmetic sequence has $a_1 = 7$ and $d = -3$. Find $a_6$ and $a_{20}$.
  • Solve $a_6 = 7 + (6-1)(-3) = 7 - 15 = -8$ $a_{20} = 7 + (20-1)(-3) = 7 - 57 = -50$
  • CheckStarting at 7 and subtracting 3 five times: $7, 4, 1, -2, -5, -8$. ✓
Answer: $a_6 = -8$ and $a_{20} = -50$.
Example 11.5.2
Writing the formula from two terms
  • ProblemAn arithmetic sequence has $a_3 = 15$ and $a_7 = 31$. Write the explicit formula.
  • ThinkFrom term 3 to term 7 is $7 - 3 = 4$ steps. Use this to find $d$, then find $a_1$.
  • Solve $a_7 = a_3 + 4d \implies 31 = 15 + 4d \implies d = 4$ $a_3 = a_1 + 2d \implies 15 = a_1 + 8 \implies a_1 = 7$ $a_n = 7 + (n-1)(4) = 4n + 3$
  • Check$a_3 = 4(3)+3 = 15$ ✓ and $a_7 = 4(7)+3 = 31$ ✓.
Answer: $a_n = 4n + 3$.
Practice 11.5Arithmetic Sequences
  1. Find the first five terms: $a_1 = 6$, $d = 4$.
  2. Find the first five terms: $a_1 = 50$, $d = -7$.
  3. Write the explicit formula: $11, 15, 19, 23, 27, \ldots$
  4. Write the explicit formula: $100, 93, 86, 79, 72, \ldots$
  5. $a_1 = 3$, $a_{10} = 48$. Find $d$ and the explicit formula.
  6. $a_4 = 22$, $a_9 = 47$. Find $a_1$ and write the explicit formula.
  7. Is $1, 4, 9, 16, 25$ arithmetic? Explain why or why not.
  8. A staircase descends at a constant rate. Step 1 is 8 inches below ground level; each step descends 7 more inches. Write an explicit formula and find the depth of step 12.
Section 11.6

Arithmetic Series

A series is the sum of the terms of a sequence. An arithmetic series is the sum of the terms of an arithmetic sequence. The sum of the first $n$ terms is called a partial sum, written $S_n$:

Arithmetic Series — Partial Sum Formulas
$$S_n = \frac{n}{2}(a_1 + a_n)$$

Use when you know both the first and last terms.

$$S_n = \frac{n}{2}\bigl[2a_1 + (n-1)d\bigr]$$

Use when you know the first term and common difference.

The Gauss Pairing Argument

To add $1 + 2 + 3 + \cdots + 100$, pair the first and last terms: $1 + 100 = 101$. Pair the second and second-to-last: $2 + 99 = 101$. There are 50 such pairs, so the sum is $50 \times 101 = 5050$. In general, $n$ terms yield $\frac{n}{2}$ pairs each summing to $a_1 + a_n$, giving $S_n = \frac{n}{2}(a_1 + a_n)$.

Example 11.6.1
Sum of the first $n$ terms
  • ProblemFind the sum of the first 10 terms of the arithmetic sequence with $a_1 = 4$ and $d = 3$.
  • Solve $a_{10} = 4 + (10-1)(3) = 31$ $S_{10} = \frac{10}{2}(4 + 31) = 5 \times 35 = 175$
  • CheckThe average value of the terms is $\frac{4+31}{2} = 17.5$. Ten terms with average 17.5 gives $10 \times 17.5 = 175$. ✓
Answer: $S_{10} = 175$.
Example 11.6.2
Finding the number of terms for a given sum
  • ProblemHow many terms of $5, 9, 13, 17, \ldots$ are needed for the partial sum to reach 285?
  • ThinkSet $S_n = 285$ with $a_1 = 5$ and $d = 4$, then solve for $n$.
  • Solve $\frac{n}{2}[2(5) + (n-1)(4)] = 285$ $\frac{n}{2}(4n + 6) = 285 \implies 2n^2 + 3n - 285 = 0$ Check $n = 11$: $S_{11} = \frac{11}{2}(50) = 275$ Check $n = 12$: $S_{12} = 6(54) = 324$
Answer: No whole number of terms gives a sum of exactly 285. The sum jumps from 275 to 324.
Student Warning — Always Check That $n$ Is a Positive Integer

Just because you set up an equation does not mean its solution is a valid index. Always verify that your answer for $n$ is a positive whole number. If it is not, the partial sum never reaches the target exactly.

Practice 11.6Arithmetic Series
  1. Find $S_{15}$: $a_1 = 2$, $d = 5$.
  2. Find $S_{20}$: $a_1 = 100$, $d = -3$.
  3. $a_1 = 7$, $a_n = 49$, $n = 11$. Find $S_{11}$.
  4. Find the sum of integers from 1 to 50.
  5. Find the sum of integers from 1 to 100.
  6. A savings plan deposits $200 the first month, then $25 more each month. How much total is deposited over 12 months?
  7. A triangular arrangement has 1 tile in row 1, 2 in row 2, …, 10 in row 10. Write this as an arithmetic series and find the total tiles.
Sections 11.7 – 11.9
Geometric Sequences and Series
Common Ratio · Explicit Formula · Finite Sums · Infinite Convergence
Section 11.7

Geometric Sequences

A geometric sequence is a sequence in which each term is obtained by multiplying the previous term by a fixed number. That fixed number is called the common ratio, denoted $r$. To check whether a sequence is geometric, divide each term by the term before it; if the ratio is constant, the sequence is geometric.

Common Ratio and Explicit Formula

$r = \dfrac{a_2}{a_1} = \dfrac{a_3}{a_2} = \dfrac{a_4}{a_3} = \cdots$

Explicit formula for a geometric sequence:

$$a_n = a_1 \cdot r^{n-1}$$

To reach term $n$, start at $a_1$ and multiply by $r$ exactly $(n-1)$ times.

TypeExplicit FormulaOperationFunction Family
Arithmetic$a_n = a_1 + (n-1)d$Add $d$ repeatedlyLinear
Geometric$a_n = a_1 \cdot r^{n-1}$Multiply by $r$ repeatedlyExponential
Example 11.7.1
Finding terms of a geometric sequence
  • Problem$a_1 = 5$ and $r = 2$. Find $a_4$ and $a_8$.
  • Solve $a_4 = 5 \cdot 2^{4-1} = 5 \cdot 8 = 40$ $a_8 = 5 \cdot 2^{7} = 5 \cdot 128 = 640$
  • CheckThe sequence: $5, 10, 20, 40, 80, 160, 320, 640$. ✓
Answer: $a_4 = 40$ and $a_8 = 640$.
Example 11.7.2
Finding the common ratio from two terms
  • ProblemA geometric sequence has $a_1 = 3$ and $a_5 = 48$. Find $r$.
  • Solve $a_5 = a_1 \cdot r^4 \implies 48 = 3 \cdot r^4 \implies r^4 = 16 \implies r = \pm 2$ If positive terms: $r = 2$, giving $3, 6, 12, 24, 48$. ✓ If alternating: $r = -2$, giving $3, -6, 12, -24, 48$.
Answer: $r = 2$ (positive terms) or $r = -2$ (alternating signs).
Example 11.7.3
A decay sequence
  • Problem$a_1 = 200$ and $r = \frac{1}{2}$. List the first five terms and write the explicit formula.
  • Solve $200, \; 100, \; 50, \; 25, \; 12.5$ $a_n = 200 \cdot \left(\dfrac{1}{2}\right)^{n-1}$
Answer: $200, 100, 50, 25, 12.5, \ldots$ with $a_n = 200 \cdot \left(\frac{1}{2}\right)^{n-1}$.
Practice 11.7Geometric Sequences
  1. First five terms: $a_1 = 4$, $r = 3$.
  2. First five terms: $a_1 = 160$, $r = \frac{1}{4}$.
  3. Write explicit formula: $2, 10, 50, 250, \ldots$
  4. Write explicit formula: $81, 27, 9, 3, 1, \ldots$
  5. $a_1 = 6$, $a_6 = 192$. Find $r$ and the explicit formula.
  6. What happens to terms when $r > 1$? When $0 < r < 1$? When $r < 0$?
  7. A population starts at 500 and doubles every hour. Write a geometric sequence and find the population after 8 hours.
Section 11.8

Geometric Series

A geometric series is the sum of the terms of a geometric sequence. The derivation of the partial sum formula is elegant: write the sum, multiply by $r$, subtract the two expressions, and solve.

Geometric Series — Partial Sum Formula
$$S_n = a_1 \cdot \frac{1 - r^n}{1 - r}, \quad r \neq 1$$

If $r = 1$, every term equals $a_1$ and $S_n = n \cdot a_1$.

Example 11.8.1
Sum of a finite geometric series — growth
  • ProblemFind the sum of the first 6 terms of the geometric sequence with $a_1 = 3$ and $r = 2$.
  • Solve $S_6 = 3 \cdot \frac{1 - 2^6}{1 - 2} = 3 \cdot \frac{1 - 64}{-1} = 3 \cdot 63 = 189$
  • CheckTerms: $3, 6, 12, 24, 48, 96$. Sum: $3+6+12+24+48+96 = 189$. ✓
Answer: $S_6 = 189$.
Example 11.8.2
Sum of a finite geometric series — decay
  • ProblemFind the sum of the first 5 terms of the geometric sequence with $a_1 = 80$ and $r = \frac{1}{2}$.
  • Solve $S_5 = 80 \cdot \frac{1 - \left(\frac{1}{2}\right)^5}{1 - \frac{1}{2}} = 80 \cdot \frac{\frac{31}{32}}{\frac{1}{2}} = 80 \cdot \frac{31}{16} = 155$
  • CheckTerms: $80, 40, 20, 10, 5$. Sum: $80+40+20+10+5 = 155$. ✓
Answer: $S_5 = 155$.
Practice 11.8Geometric Series
  1. $S_5$: $a_1 = 2$, $r = 4$.
  2. $S_8$: $a_1 = 5$, $r = 3$.
  3. $S_6$: $a_1 = 100$, $r = \frac{1}{10}$.
  4. $S_5$: $a_1 = 7$, $r = -2$.
  5. A city's population was 50,000 in 2015 and grew 8% per year. Write the population series for 2015–2020 and find the combined total population over those six years.
Section 11.9

Finite and Infinite Series

A finite series has a specific number of terms and always has a definite numerical value. An infinite series continues without stopping. At first, summing infinitely many terms seems impossible. But when the terms of a geometric series shrink fast enough, the infinite sum converges to a finite value.

Consider the geometric series with $a_1 = 1$ and $r = \frac{1}{2}$: $1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots$ The partial sums are $S_1 = 1$, $S_2 = 1.5$, $S_3 = 1.75$, $S_4 = 1.875$, $S_5 = 1.9375, \ldots$ approaching 2. When $|r| < 1$, the factor $r^n \to 0$ as $n \to \infty$, and the partial sum formula simplifies to a finite value.

Infinite Geometric Series — Sum Formula
$$S = \frac{a_1}{1 - r}, \quad \text{when } |r| < 1$$

When $|r| \geq 1$, the series diverges — the terms do not shrink to zero and the sum has no finite value.

Example 11.9.1
Sum of an infinite geometric series
  • ProblemFind the sum of the infinite geometric series with $a_1 = 12$ and $r = \frac{1}{3}$.
  • ThinkSince $|r| = \frac{1}{3} < 1$, the series converges.
  • Solve $S = \frac{12}{1 - \frac{1}{3}} = \frac{12}{\frac{2}{3}} = 12 \cdot \frac{3}{2} = 18$
  • Check$S_1 = 12$, $S_2 = 16$, $S_3 \approx 17.33$, $S_4 \approx 17.78$, approaching 18. ✓
Answer: $S = 18$.
Example 11.9.2
Recognizing a divergent series
  • ProblemDoes the infinite geometric series with $a_1 = 5$ and $r = 1.2$ converge?
  • Solve$|r| = 1.2 > 1$. The terms grow without bound. The series diverges.
Answer: No. The series diverges — there is no finite sum.
A Preview of Calculus

The idea that an infinite sum can equal a finite number is one of the most powerful and counterintuitive ideas in mathematics. In Calculus, this is generalized to power series, Taylor series, and Maclaurin series — tools that allow approximation of any function with a polynomial. The convergence condition $|r| < 1$ is your first encounter with the concept of a limit.

Practice 11.9Finite and Infinite Series
  1. $S$: $a_1 = 8$, $r = \frac{1}{4}$.
  2. $S$ of $6 + 2 + \frac{2}{3} + \frac{2}{9} + \cdots$
  3. Does $3 + 6 + 12 + 24 + \cdots$ converge? Explain.
  4. Does $10 - 5 + 2.5 - 1.25 + \cdots$ converge? If so, find the sum.
  5. A ball dropped from 10 m rises to 60% of its previous height after each bounce. Find the total upward distance using the infinite geometric series formula.
Sections 11.10 – 11.12
Sigma Notation, Finance, and Real-World Applications
Sigma Notation · Compound Interest · Annuities · Science and Computing
Section 11.10

Sigma Notation

Sigma notation (also called summation notation) uses the capital Greek letter $\Sigma$ (sigma) to write a series compactly. The index of summation is the variable that changes at each step. The lower limit tells where the sum starts. The upper limit tells where it ends.

Reading Sigma Notation
$$\sum_{k=m}^{n} a_k = a_m + a_{m+1} + a_{m+2} + \cdots + a_n$$

"The sum from $k$ equals $m$ to $n$ of $a_k$." The index letter does not affect the result — $\sum_{k=1}^{5} 2k$ and $\sum_{j=1}^{5} 2j$ produce the same value.

Example 11.10.1
Evaluating a sigma notation sum
  • ProblemEvaluate $\displaystyle\sum_{k=1}^{5}(2k+1)$.
  • Solve $k=1: 3, \quad k=2: 5, \quad k=3: 7, \quad k=4: 9, \quad k=5: 11$ $3 + 5 + 7 + 9 + 11 = 35$
Answer: 35.
Example 11.10.2
Writing a series in sigma notation
  • ProblemWrite $3 + 6 + 12 + 24 + 48$ in sigma notation.
  • ThinkGeometric series: $a_1 = 3$, $r = 2$, general term $= 3 \cdot 2^{k-1}$. Five terms, so $k$ runs from 1 to 5.
  • Solve $\displaystyle\sum_{k=1}^{5} 3 \cdot 2^{k-1}$
  • Check$k=1$: $3 \cdot 1 = 3$ ✓; $k=3$: $3 \cdot 4 = 12$ ✓; $k=5$: $3 \cdot 16 = 48$ ✓.
Answer: $\displaystyle\sum_{k=1}^{5} 3 \cdot 2^{k-1}$.
Example 11.10.3
Sigma notation with a different lower limit
  • ProblemEvaluate $\displaystyle\sum_{k=3}^{6} k^2$.
  • Solve $k=3: 9, \quad k=4: 16, \quad k=5: 25, \quad k=6: 36$ $9 + 16 + 25 + 36 = 86$
Answer: 86.

Useful Sigma Identities

IdentityFormula
Sum of a constant$\displaystyle\sum_{k=1}^{n} c = cn$
Sum of first $n$ integers$\displaystyle\sum_{k=1}^{n} k = \dfrac{n(n+1)}{2}$
Sum of squares$\displaystyle\sum_{k=1}^{n} k^2 = \dfrac{n(n+1)(2n+1)}{6}$
Practice 11.10Sigma Notation
  1. Evaluate $\displaystyle\sum_{k=1}^{4}(3k-1)$.
  2. Evaluate $\displaystyle\sum_{k=0}^{5} 2^k$.
  3. Evaluate $\displaystyle\sum_{k=2}^{6} k$.
  4. Write $5+8+11+14+17$ in sigma notation.
  5. Write $1+4+9+16+25+36$ in sigma notation.
  6. Write $10+5+2.5+1.25+\cdots$ (infinite) in sigma notation.
  7. A store's daily sales for 7 days follow $a_k = 200 + 50k$ dollars. Write in sigma notation and evaluate.
Section 11.11

Compound Interest and Geometric Patterns

When money is invested at a fixed interest rate and the interest is compounded, the account balance grows geometrically. Each period, the balance is multiplied by the common ratio $1 + r$ (where $r$ is the periodic rate). Recall from Chapter 9 that the compound interest formula for annual compounding is:

Compound Interest as a Geometric Sequence
$$A = P(1 + r)^t$$

$P$ = principal; $r$ = annual rate (decimal); $t$ = time in years. This is a geometric sequence with $a_1 = P(1+r)$ and common ratio $1+r$.

Example 11.11.1
Compound interest as a geometric sequence
  • Problem$\$2,000$ invested at 6% annual interest, compounded annually. Write the first five annual balances and find the balance at year 10.
  • Solve $a_1 = 2000(1.06) \approx \$2120, \quad a_2 \approx \$2247.20$ $a_3 \approx \$2382.03, \quad a_4 \approx \$2524.95, \quad a_5 \approx \$2675.25$ $A_{10} = 2000 \cdot (1.06)^{10} \approx \$3{,}581.70$
  • CheckEach term is 1.06 times the previous one — consistent geometric behavior. ✓
Answer: After 10 years, the account holds approximately $\$3,581.70$.

Annuities as Geometric Series

An annuity is a series of equal payments made at regular intervals. If each payment earns interest, the total value is the sum of a geometric series. For $n$ end-of-year payments of amount $PMT$ at rate $r$, the future value is:

Future Value of an Annuity
$$FV = PMT \cdot \frac{(1+r)^n - 1}{r}$$

This is the sum of a geometric series with $a_1 = PMT$, ratio $= 1+r$, and $n$ terms.

Example 11.11.2
Future value of an annuity
  • ProblemYou deposit $\$1,000$ at the end of each year for 4 years into an account earning 8% annual interest. What is the total value at the end of year 4?
  • ThinkEach deposit earns interest for the remaining years. The year-4 deposit earns nothing; the year-3 deposit earns one year of interest; and so on.
  • Solve $1000 + 1000(1.08) + 1000(1.08)^2 + 1000(1.08)^3$ $= 1000 + 1080 + 1166.40 + 1259.71 \approx \$4{,}506.11$
  • CheckUsing the formula: $1000 \cdot \frac{(1.08)^4 - 1}{0.08} \approx \$4,506.11$ ✓
Answer: Approximately $\$4,506.11$.
MGU Library Connection
  • Annuities, present value, future value, and amortization — MGU Finance course
  • Compound interest and exponential models — Algebra II Chapter 9: Exponential Functions
  • Geometric series and limits — Precalculus and Calculus I
  • Investment and economics applications — MGU Economics and Investment modules
Practice 11.11Compound Interest and Annuities
  1. $\$5,000$ is invested at 4% annual interest, compounded annually. Write the balance as a geometric sequence for years 1–5. What is the balance at the end of year 20?
  2. You deposit $\$200$ at the end of each year for 5 years into an account earning 6% annual interest. Find the total value at the end of year 5.
  3. Explain in your own words why compound interest is geometric and not arithmetic.
Section 11.12

Sequences in Science and Computing

Sequences and series appear far beyond finance. Biology, physics, chemistry, and computer science all use the tools developed in this chapter.

Biology: Population Models

A population that multiplies by a fixed growth factor each generation follows a geometric sequence. If a colony starts with 50 organisms and triples each generation: $a_n = 50 \cdot 3^{n-1}$. More realistic models use differential equations in Calculus, but geometric sequences provide the foundation.

Physics: Bouncing Ball

A ball dropped from height $h$ bounces to a fraction $r$ of that height after each bounce ($0 < r < 1$). The heights form a geometric sequence $h, hr, hr^2, \ldots$ The total upward distance traveled over all bounces is an infinite geometric series.

Chemistry: Radioactive Decay

After each half-life, half the original material remains. The amounts form a geometric sequence with $r = \frac{1}{2}$: after $n$ half-lives, the amount is $a_n = a_0 \cdot \left(\frac{1}{2}\right)^n$.

Computer Science: Binary Search and Binary Numbers

Binary search halves the number of remaining candidates at each step, producing geometric decay in the search space. Starting with $n$ elements, the algorithm terminates in at most $\log_2 n$ steps — a direct consequence of the geometric structure. Binary numbers themselves are finite geometric sums: the binary number $1011_2$ represents $1 \cdot 2^3 + 0 \cdot 2^2 + 1 \cdot 2^1 + 1 \cdot 2^0 = 11$.

Example 11.12.1
Radioactive decay
  • ProblemA radioactive substance has a half-life of 10 years. A sample starts with 640 grams. How much remains after 50 years?
  • ThinkAfter 50 years, there are $50 \div 10 = 5$ half-lives. Each halves the amount.
  • Solve $a_5 = 640 \cdot \left(\frac{1}{2}\right)^5 = 640 \cdot \frac{1}{32} = 20 \text{ grams}$
Answer: 20 grams remain after 50 years.
Practice 11.12Science and Computing Applications
  1. A population of deer doubles every 6 years. Starting at 300, write a geometric sequence for the population every 6 years over 30 years, and find the population at the end of 30 years.
  2. A bouncing ball is dropped from 5 meters and rises to 70% of its previous height after each bounce. Find the total upward distance the ball travels over all bounces.
  3. A binary search algorithm starts with 1024 elements. How many steps are needed to reduce the search to 1 element?
  4. The binary number $1101_2$ represents $1 \cdot 2^3 + 1 \cdot 2^2 + 0 \cdot 2^1 + 1 \cdot 2^0$. Evaluate this sum. What decimal number does it equal?
  5. A factory produces 20 items on day 1, 22 on day 2, 24 on day 3, and so on (arithmetic growth of 2 per day). Write the series for the first 10 days and find the total output.
Sections 11.13 – 11.14
Common Mistakes and the Road Ahead
Seven Recurring Errors · Connections to Precalculus, Calculus, and Beyond
Section 11.13

Common Sequence and Series Mistakes

Mistake 1 — Confusing Index with Value
For the sequence $3, 7, 11, 15$: $a_4 = 11$
Correct: $a_3 = 11$ (third position) and $a_4 = 15$ (fourth position). The index is position; the value is the term at that position. These are not the same.
Mistake 2 — Using $a_n = a_1 + nd$ Instead of $a_n = a_1 + (n-1)d$
With $a_1 = 3$, $d = 4$: $a_1 = 3 + 1(4) = 7$
Correct: From $a_1$ to $a_n$ there are $(n-1)$ steps. Use $a_n = a_1 + (n-1)d$. Always verify at $n = 1$: the formula should return $a_1$.
Mistake 3 — Writing $a_n = a_1 \cdot r^n$ Instead of $a_n = a_1 \cdot r^{n-1}$
With $a_1 = 2$, $r = 3$: $a_1 = 2 \cdot 3^1 = 6$
Correct: The formula must satisfy $a_1 = a_1 \cdot r^{1-1} = a_1 \cdot r^0 = a_1$. Use $a_n = a_1 \cdot r^{n-1}$.
Mistake 4 — Applying the Infinite Sum Formula When $|r| \geq 1$
Series $3 + 6 + 12 + 24 + \cdots$: $S = \frac{3}{1-2} = -3$
Correct: The infinite sum formula $S = \frac{a_1}{1-r}$ only works when $|r| < 1$. Here $r = 2 > 1$; the series diverges and has no finite sum. Always check $|r|$ before applying the formula.
Mistake 5 — Mixing Up $n$ (Count of Terms) with $a_n$ (Last Term)
In $S_n = \frac{n}{2}(a_1 + a_n)$, substituting the value of $a_n$ in place of $n$
Correct: $n$ is the count of terms (a counting number like 10 or 20). $a_n$ is the value of the last term. These are different quantities. Label each before substituting.
Mistake 6 — Assuming Every Sequence Is Arithmetic or Geometric
Sequence $1, 4, 9, 16, 25$: common difference is $d = 3$
Correct: The differences are $3, 5, 7, 9$ — not constant. The ratios are $4, 2.25, 1.78$ — not constant. This is the sequence of perfect squares and is neither arithmetic nor geometric. Always check before classifying.
Mistake 7 — Incorrect Sigma Notation (Formula Doesn't Match the Terms)
Sum of squares of even numbers $2, 4, 6, 8, 10$: $\displaystyle\sum_{k=1}^{5} k^2$
Correct: $\displaystyle\sum_{k=1}^{5} (2k)^2$. The expression inside the sigma must generate the actual terms. Check each: when $k=1$, $(2 \cdot 1)^2 = 4$ ✓.
Practice 11.13Error Analysis

Identify the mistake in each problem and write the correct solution.

  1. A student says $5, 10, 15, 20$ is geometric with $r = 5$ because $a_2 - a_1 = 5$. What is wrong?
  2. A student computes $a_8$ for $a_1 = 3$, $d = 4$ as $a_8 = 3 + 8(4) = 35$. What is the error?
  3. A student evaluates $\displaystyle\sum_{k=1}^{4} k^2$ as $1 + 2 + 3 + 4 = 10$. What went wrong?
  4. A student applies the infinite sum formula to $2 + 4 + 8 + 16 + \cdots$ and concludes $S = \frac{2}{1-2} = -2$. What is wrong?
  5. A student writes the explicit formula for $3, 7, 11, 15, \ldots$ as $a_n = 3 + 4n$. Is this correct? Verify at $n = 1$ and correct if needed.
Section 11.14

Preparing for Advanced Mathematics

Sequences and series connect to many areas of advanced mathematics studied beyond Algebra II.

In Precalculus, arithmetic and geometric sequences reappear alongside mathematical induction — a proof technique used to establish formulas that hold for all positive integers. The sigma notation introduced here becomes the foundation for a deeper study of summation formulas and their derivations.

In Calculus, infinite series are among the most powerful tools in mathematics. The idea that an infinite sum can converge to a finite value is generalized to power series, Taylor series, and Maclaurin series. These allow Calculus to approximate any function with polynomials — the foundation of numerical computing, physics simulations, and engineering analysis. The partial sum $S_n$ is an early version of the concept of a limit: as $n$ grows without bound, $S_n$ approaches a fixed value.

In Discrete Mathematics, recursive formulas, sigma notation, counting, and mathematical induction are all central topics. In Statistics, the arithmetic of sums underpins expected value, variance, and regression formulas. In Finance and Economics, geometric series are the mathematical foundation of compound interest, annuities, present value, and loan amortization. In Computer Science, recursive algorithms, dynamic programming, and complexity analysis (Big-O notation) all draw on the sequence and series ideas in this chapter.

Looking Ahead to Chapter 12: Conic Sections

Chapter 11 described patterns that evolve in discrete steps — arithmetic (additive) and geometric (multiplicative). Chapter 12 turns to a different domain: the geometry of curves. Conic sections — circles, parabolas, ellipses, and hyperbolas — arise when a plane cuts through a cone and appear in orbital mechanics, engineering, optics, and architecture. The algebra needed for Chapter 12 draws on completing the square (Chapter 3), the distance formula, and coordinate graphing. What is new is the precise connection between geometric definitions involving distances and the algebraic equations that describe them.

Chapter 11 — Back Matter

Chapter Summary

Key Terms Review

sequence
An ordered list of numbers that follows a rule; a function whose domain is the positive integers.
term
A single number in a sequence, occupying a specific position.
index
The position number of a term in a sequence, written as a subscript: $a_n$ has index $n$.
explicit formula
A formula giving any term of a sequence directly from the index, without requiring prior terms.
recursive formula
A formula defining each term from one or more preceding terms, together with a starting value.
arithmetic sequence
A sequence in which each term is obtained by adding a fixed common difference to the previous term.
common difference
The fixed number $d$ added at each step of an arithmetic sequence.
arithmetic series
The sum of the terms of an arithmetic sequence.
partial sum
$S_n$: the sum of the first $n$ terms of a sequence.
geometric sequence
A sequence in which each term is obtained by multiplying the previous term by a fixed common ratio.
common ratio
The fixed multiplier $r$ at each step of a geometric sequence.
geometric series
The sum of the terms of a geometric sequence.
finite series
A series with a specific number of terms, always producing a definite numerical value.
infinite series
A series that continues without end; may converge or diverge.
convergent series
An infinite series whose partial sums approach a finite limit. Infinite geometric series converge when $|r| < 1$.
divergent series
An infinite series whose partial sums grow without bound; no finite sum exists.
sigma notation
$\sum_{k=m}^{n} a_k$ — compact notation for a series using the Greek letter $\Sigma$ (sigma) for summation.
annuity
A series of equal payments at regular intervals; its total value is a geometric series sum.

Skill Practice

Part AArithmetic Sequences
  1. First five terms: $a_1 = -3$, $d = 6$.
  2. 25th term: $a_1 = 10$, $d = -2$.
  3. Explicit formula: $7, 13, 19, 25, 31, \ldots$
  4. $a_5 = 30$, $a_{12} = 65$. Find $a_1$ and $d$.
  5. How many terms in $3, 7, 11, \ldots, 103$?
Part BGeometric Sequences
  1. First five terms: $a_1 = 2$, $r = -3$.
  2. 7th term: $a_1 = 1$, $r = \frac{1}{3}$.
  3. Explicit formula: $4, 12, 36, 108, \ldots$
  4. $a_2 = 10$, $a_5 = 1250$. Find $r$ and $a_1$.
  5. Is $2, -4, 8, -16, 32$ geometric? If so, find $r$.
Part CSeries
  1. $S_{15}$: $a_1 = 4$, $d = 6$.
  2. Sum of first 20 terms of $50, 45, 40, 35, \ldots$
  3. $S_7$: $a_1 = 3$, $r = 2$.
  4. Sum of first 8 terms of $1, 5, 25, 125, \ldots$
  5. Infinite sum: $a_1 = 15$, $r = \frac{2}{5}$.
  6. Does $4 + 6 + 9 + 13.5 + \cdots$ converge? Explain.
Part DSigma Notation
  1. Evaluate $\displaystyle\sum_{k=1}^{6}(4k-3)$.
  2. Evaluate $\displaystyle\sum_{k=0}^{4} 3 \cdot 2^k$.
  3. Write $10+13+16+19+22$ in sigma notation.
  4. Write $1 + \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \frac{1}{81}$ in sigma notation.

Word Problems

  1. A city's population in 2020 was 40,000 and grows by 1,200 people per year. Write an explicit formula for the population in year $n$ (where $n = 1$ is 2020). What is the expected population in 2035?
  2. A bacterial population triples every 4 hours, starting with 100 bacteria. Write an explicit formula for the population after $n$ four-hour periods. How large is the population after 24 hours?
  3. A student deposits $\$500$ per year into an account earning 7% annually. Using the geometric series approach, find the total value of the account at the end of 6 years.
  4. A ball is dropped from 12 meters. After each bounce, it rises to 55% of its previous height. Find the total upward distance traveled over all bounces.
  5. A sequence of salaries over a 5-year contract starts at $\$48,000$ and increases by $\$3,500$ per year. Find the total earnings over the full 5-year contract.
  6. A savings plan deposits $\$300$ the first month, then $\$20$ more each month. How much total is deposited over the first 18 months?

Error Analysis

A student made an error in each problem. Identify it and write the correct solution.

Problem 1
$a_1 = 5$, $d = 3$: the student writes $a_n = 5 + 3n$ and finds $a_1 = 5 + 3(1) = 8$.

What is the error? What is the correct formula and value of $a_1$?

Problem 2
$a_1 = 4$, $r = 2$: the student writes $a_5 = 4 \cdot 2^5 = 128$.

What is the error? What is the correct value of $a_5$?

Problem 3
A student evaluates $\displaystyle\sum_{k=1}^{3} k^2$ as $1 + 2 + 3 = 6$.

Identify all errors. What is the correct value?

Problem 4
A student applies the infinite sum formula to $5 + 10 + 20 + 40 + \cdots$ and writes $S = \frac{5}{1-2} = -5$.

What is the error? What is the correct conclusion?

Chapter 11 Checkpoint

Chapter 11 Checkpoint
Complete without assistance — assesses all key skills and concepts from Chapter 11
Part A — Identify and Classify
  1. Is the sequence $3, 6, 12, 24, 48$ arithmetic, geometric, or neither?
  2. Is the sequence $10, 7, 4, 1, -2$ arithmetic, geometric, or neither?
  3. Is the sequence $1, 3, 6, 10, 15$ arithmetic, geometric, or neither?
Part B — Formulas
  1. Write the explicit formula for the arithmetic sequence with $a_1 = 8$ and $d = -4$. Find $a_{15}$.
  2. Write the explicit formula for the geometric sequence with $a_1 = 6$ and $r = \frac{1}{3}$. Find $a_5$.
  3. Write a recursive formula for the sequence $1, 5, 25, 125, \ldots$
Part C — Series
  1. Find the sum of the first 12 terms of the arithmetic sequence with $a_1 = 3$ and $d = 5$.
  2. Find the sum of the first 7 terms of the geometric sequence with $a_1 = 1$ and $r = 4$.
  3. Determine whether the infinite geometric series $18 + 12 + 8 + \frac{16}{3} + \cdots$ converges. If it does, find its sum.
Part D — Sigma Notation
  1. Evaluate $\displaystyle\sum_{k=1}^{5}(2k+3)$.
  2. Write $2 + 6 + 18 + 54 + 162$ in sigma notation.
Part E — Applications
  1. A savings account starts with $\$1,500$ and earns 4% annual interest, compounded annually. How much money is in the account after 10 years?
  2. A theater adds 4 seats per row, starting with 20 seats in row 1. Find the total number of seats in the first 8 rows.
  3. A bouncing ball is dropped from 9 meters. After each bounce, it returns to $\frac{2}{3}$ of its previous height. What is the total upward distance the ball travels over all bounces?
Part F — Error Analysis
  1. A student computes $S_{10}$ for an arithmetic sequence with $a_1 = 5$ and $d = 3$, finding $a_{10} = 5 + 10(3) = 35$, then $S_{10} = \frac{10}{2}(5 + 35) = 200$. Identify the error in finding $a_{10}$. What is the correct value of $S_{10}$?
  2. A student claims that the infinite series $7 + 7 + 7 + \cdots$ with $r = 1$ converges to $\frac{7}{1-1}$. Explain the error clearly.
Looking Ahead
Chapter 12: Conic Sections

Chapter 11 described patterns that evolve over discrete steps — one term to the next, building by addition or multiplication in a predictable way. The functions involved were arithmetic (linear) and geometric (exponential) in structure, and they connected directly to financial, biological, and computational models.

Chapter 12 turns to a completely different domain: the geometry of curves. Conic sections — circles, parabolas, ellipses, and hyperbolas — arise whenever a plane cuts through a cone. They appear in physics (the orbits of planets and comets follow ellipses and parabolas), in engineering (parabolic reflectors focus radio signals and light), in architecture (arches take elliptical or parabolic shapes), and in optics (lenses and mirrors are conic cross-sections). Each curve is defined by a precise geometric relationship involving distances, and each has an algebraic equation that captures that relationship exactly. Chapter 12 builds that connection carefully, starting with circles and expanding to ellipses, hyperbolas, and the parabola revisited as a conic.

MGU Library Connection
  • Sequence and Series Formulas — Appendix J of this textbook
  • Exponential and Logarithmic Rules — Appendix I (connection to geometric sequences)
  • Compound Interest Reference — MGU Finance course and Chapter 9 of this textbook
  • Precalculus: Mathematical Induction and Advanced Series
  • Calculus I: Limits, Infinite Series, and the Integral as a Sum
  • Discrete Mathematics: Recursion, Generating Functions, Counting
  • MGU Dictionary entries: sequence, series, arithmetic sequence, geometric sequence, sigma notation, convergence