MGU Mathematics Series — Trigonometry and Precalculus
Part III: Advanced Algebraic Functions
Sequences · Series · Sigma Notation · Mathematical Induction
Chapter 10 of 14
MGU Mathematics Series  ·  Trigonometry and Precalculus  ·  Library Textbook Edition

Chapter 10:
Sequences, Series,
and Mathematical Induction

Discrete functions, arithmetic and geometric sequences, finite and infinite series, sigma notation, partial sums, and mathematical induction — the mathematics of step-by-step processes and the gateway to infinite analysis

■ Math Level: Trigonometry & Precalculus ■

■ 14 Sections  •  Part III of 5 ■

Follows Chapter 9: Exponential and Logarithmic Functions  ·  Prepares students for Calculus I, Calculus II, Discrete Mathematics, and Computer Science

Chapter 10 — Front Matter

Purpose and Overview

Mathematics often studies how things change one step at a time. A population grows year by year. A loan balance decreases with each monthly payment. A bouncing ball reaches lower and lower heights with each bounce. An algorithm processes one more data point with each iteration. These are not continuous processes — they are step-by-step processes, and the mathematical tools for describing them are sequences and series.

Chapter 10 extends the work on sequences and series that students began in Algebra II and builds toward calculus-level thinking. A sequence is a list of numbers generated by a rule. A series is the running total of those numbers. Together, they model discrete processes in finance, computing, physics, biology, engineering, and data science. This chapter also introduces mathematical induction — one of the most important proof techniques in all of mathematics — which allows you to prove that a statement is true for every positive integer by a logical argument that links each case to the one before it.

Students should leave this chapter understanding that sequences are functions with whole-number inputs, that series accumulate values, that some infinite series converge to a finite sum, and that mathematical induction provides a rigorous way to establish general truths.

Opening Question

Suppose you deposit $1,000 into a savings account that earns 5% annual interest, compounded annually. You never make another deposit. How much after 10 years? After 30 years? Now suppose instead of one deposit, you make a $1,000 deposit every year for 30 years. Each deposit earns interest for a different number of years. To find the total value of all those deposits together, you need a series. This chapter gives you the tools to answer such questions precisely — for financial models, for computing algorithms, for scientific measurements, and for infinite mathematical processes.

14Sections
13Objectives
21Key Terms
15+Worked Examples

Learning Objectives

  1. 1Define a sequence as a function with whole-number domain and write sequences using function notation.
  2. 2Write explicit formulas for sequences and use them to find specific terms.
  3. 3Write recursive formulas for sequences and generate terms step by step.
  4. 4Identify arithmetic sequences, write their explicit and recursive formulas, and find specific terms and sums.
  5. 5Identify geometric sequences, write their explicit and recursive formulas, and find specific terms and sums.
  6. 6Evaluate finite arithmetic and geometric series using summation formulas.
  7. 7Determine whether an infinite geometric series converges and find its sum when it does.
  8. 8Write and evaluate series using sigma notation.
  9. 9Find partial sums of sequences and interpret their convergence behavior.
  10. 10Recognize patterns in sequences and form reasonable conjectures.
  11. 11Use mathematical induction to prove formulas involving positive integers.
  12. 12Apply sequences and series to finance, computing, and science contexts.
  13. 13Identify and correct common mistakes with sequences, series, and induction.
Chapter 10 — Contents

Sections at a Glance

Sections 10.1 – 10.3
Sequences: Discrete Functions and Their Formulas
Sequences as Functions · Explicit Formulas · Recursive Formulas
Section 10.1

Sequences as Discrete Functions

You are already familiar with functions. A function takes an input, applies a rule, and produces an output. A sequence is a special kind of function whose inputs are restricted to the positive integers. Instead of writing f(1), f(2), f(3), …, we write a₁, a₂, a₃, … and call each value a term of the sequence. The general term aₙ describes the output when the input is the positive integer n, and n is called the index.

This definition matters because everything you know about functions applies to sequences. A sequence has a domain (the positive integers), a codomain (typically the real numbers), and a rule mapping each input to exactly one output. A sequence can be increasing, decreasing, bounded, unbounded, or oscillating. Its graph consists of isolated points, not a continuous curve, because its domain consists only of whole numbers.

Definition — Sequence

A sequence is a function a : ℤ⁺ → ℝ whose domain is the set of positive integers. The output at input n is written aₙ and called the n-th term. The index n tracks position in the list.

Graphing a Sequence

Because a sequence is a function with domain {1, 2, 3, 4, …}, its graph consists of discrete points of the form (n, aₙ). The horizontal axis represents n (the index); the vertical axis represents aₙ (the value). Points are never connected — doing so would incorrectly imply the sequence is defined between integers.

Graph Instruction — Discrete Points

Graph the sequence aₙ = 3n − 1 for n = 1 through n = 6. Plot the points (1, 2), (2, 5), (3, 8), (4, 11), (5, 14), (6, 17). Leave the points unconnected. Notice they lie along a line — this sequence grows linearly because its formula is linear in n.

Conceptual Note — Sequences vs. Functions

The function f(x) = 3x − 1 is defined for all real numbers and has a continuous graph. The sequence aₙ = 3n − 1 is defined only at positive integers and has an isolated-point graph. Although they share the same formula, they are different mathematical objects. In calculus, this distinction becomes significant when studying limits of sequences versus limits of functions.

Section 10.2

Explicit Formulas

An explicit formula gives aₙ directly as a function of n. You substitute the index n and immediately compute the term — without needing any previous terms. This is the most computationally powerful form because it allows you to jump directly to any term.

Example 10.2.1
Finding Terms of a Sequence from an Explicit Formula
  • ProblemLet aₙ = n² − n + 1. Find a₁, a₄, a₁₀, and a₁₀₀.
  • ThinkSubstitute each value of n directly into the formula. The explicit form lets us jump to any term immediately.
  • Solve a₁ = 1² − 1 + 1 = 1 a₄ = 4² − 4 + 1 = 16 − 4 + 1 = 13 a₁₀ = 10² − 10 + 1 = 100 − 10 + 1 = 91 a₁₀₀ = 100² − 100 + 1 = 10000 − 100 + 1 = 9901
  • CheckValues grow as n grows, which makes sense since n² dominates.
Answer: a₁ = 1, a₄ = 13, a₁₀ = 91, a₁₀₀ = 9901
Example 10.2.2
Writing an Explicit Formula from a Pattern
  • ProblemWrite an explicit formula for the sequence 3, 6, 12, 24, 48, …
  • ThinkEach term is twice the previous. The first term is 3. The pattern is multiplying by 2 at each step, so the exponent should be one less than the index.
  • RepresentTerms are 3, 3·2, 3·2², 3·2³, … The exponent is always n − 1.
  • Solveaₙ = 3 · 2^(n−1)
  • Checka₁ = 3·2⁰ = 3 ✓ a₂ = 3·2¹ = 6 ✓ a₃ = 3·2² = 12 ✓
Answer: aₙ = 3 · 2^(n−1)
Student Warning — Always Verify Your Formula

Always verify an explicit formula by substituting n = 1, n = 2, and n = 3 and checking that you get the correct terms. A formula may look right and still be off by one in the exponent or have an incorrect constant.

Section 10.3

Recursive Formulas

A recursive formula (also called a recurrence relation) defines each term in terms of one or more previous terms. It requires two parts: (1) one or more initial conditions giving explicit starting values, and (2) a recurrence rule expressing aₙ in terms of earlier terms. Recursive formulas are natural when a process is defined by how it changes at each step rather than by a direct formula for each term.

Example 10.3.1
Generating Terms from a Recursive Formula
  • ProblemA sequence is defined by a₁ = 5 and aₙ = aₙ₋₁ + 4 for n ≥ 2. Write the first five terms.
  • ThinkStart at a₁ = 5 and apply the rule repeatedly, adding 4 each time.
  • Solve a₁ = 5 a₂ = a₁ + 4 = 5 + 4 = 9 a₃ = a₂ + 4 = 9 + 4 = 13 a₄ = a₃ + 4 = 13 + 4 = 17 a₅ = a₄ + 4 = 17 + 4 = 21
Answer: 5, 9, 13, 17, 21
The Fibonacci Sequence

The most famous recursive sequence is defined by: F₁ = 1, F₂ = 1, Fₙ = Fₙ₋₁ + Fₙ₋₂ for n ≥ 3

This gives: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, … — each term is the sum of the two terms before it. The Fibonacci sequence appears in plant growth, spiral shells, branching trees, financial models, and computer algorithms. It requires two initial conditions because each term depends on two previous terms.

Connection to Computing

Recursive definitions are natural in computing. Most programming languages allow functions to call themselves — a technique called recursion. Recursive sequence definitions are the mathematical counterpart to recursive algorithms. Understanding both forms in mathematics prepares you to think recursively in code.

Sections 10.4 – 10.8
Arithmetic and Geometric Sequences and Series
Arithmetic Sequences · Arithmetic Series · Geometric Sequences · Geometric Series · Infinite Series
Section 10.4

Arithmetic Sequences

An arithmetic sequence is a sequence in which each term is obtained from the previous term by adding a fixed constant called the common difference d. If you subtract any term from the next and always get the same result, the sequence is arithmetic: d = aₙ − aₙ₋₁.

SequenceCommon DifferencePattern
3, 7, 11, 15, 19, …d = 4Increasing
10, 7, 4, 1, −2, …d = −3Decreasing
½, 1, 3/2, 2, 5/2, …d = ½Increasing
Arithmetic Sequence — Explicit Formula aₙ = a₁ + (n − 1)d

The first term is a₁, and each step increases by d. After n − 1 steps you have added d exactly n − 1 times. An arithmetic sequence corresponds to a linear function of n: plotting an arithmetic sequence gives points lying exactly on a line with slope d. Arithmetic sequences are the discrete analog of linear functions.

Example 10.4.1
Finding a Specific Term of an Arithmetic Sequence
  • ProblemAn arithmetic sequence has a₁ = 7 and d = −3. Find a₂₀.
  • Representaₙ = a₁ + (n − 1)d
  • Solvea₂₀ = 7 + (20 − 1)(−3) = 7 + (19)(−3) = 7 − 57 = −50
  • CheckThe sequence is decreasing (d < 0), so a₂₀ should be much less than 7. The value −50 is reasonable.
Answer: a₂₀ = −50
Example 10.4.2
Finding the Formula from Two Given Terms
  • ProblemAn arithmetic sequence has a₅ = 23 and a₁₂ = 58. Find the explicit formula.
  • ThinkWrite two equations using the explicit formula. Solve the system for a₁ and d.
  • Solve a₅ = a₁ + 4d = 23 ...(1) a₁₂ = a₁ + 11d = 58 ...(2) Subtract (1) from (2): 7d = 35 → d = 5 Substitute back into (1): a₁ + 4(5) = 23 → a₁ = 3 aₙ = 3 + (n − 1)(5) = 5n − 2
  • Checka₅ = 5(5) − 2 = 23 ✓ a₁₂ = 5(12) − 2 = 58 ✓
Answer: aₙ = 5n − 2
Section 10.5

Arithmetic Series

A series is the sum of the terms of a sequence. An arithmetic series is the sum of the first n terms of an arithmetic sequence. The elegant derivation of the sum formula is attributed to the young Carl Friedrich Gauss, who reportedly computed 1 + 2 + 3 + ⋯ + 100 = 5050 in seconds by recognizing that pairs of terms sum to the same value.

Gauss's Pairing Argument

Write Sₙ forward and backward, then add the two expressions term by term. Each of the n pairs sums to (a₁ + aₙ), giving 2Sₙ = n(a₁ + aₙ).

Arithmetic Series — Sum Formulas Sₙ = n/2 · (a₁ + aₙ) or Sₙ = n/2 · (2a₁ + (n−1)d)
Example 10.5.1
Finding an Arithmetic Series Sum
  • ProblemFind the sum of the first 40 terms of the arithmetic sequence 5, 9, 13, 17, …
  • Representa₁ = 5, d = 4, n = 40
  • Solve a₄₀ = 5 + 39(4) = 5 + 156 = 161 S₄₀ = 40/2 · (5 + 161) = 20 · 166 = 3320
  • CheckAverage of first and last: (5 + 161)/2 = 83. Total: 83 × 40 = 3320 ✓
Answer: S₄₀ = 3320
Example 10.5.2
Finding the Number of Terms Given the Sum
  • ProblemAn arithmetic sequence has a₁ = 3 and d = 2. How many terms are needed for the sum to equal 80?
  • ThinkSet up the sum formula equal to 80 and solve the resulting quadratic for n.
  • Solve Sₙ = n/2 · (2(3) + (n−1)(2)) = n/2 · (6 + 2n − 2) = n/2 · (4 + 2n) = n(2 + n) = n² + 2n Set n² + 2n = 80: n² + 2n − 80 = 0 (n − 8)(n + 10) = 0 n = 8 (n must be positive)
  • CheckS₈ = 8/2 · (2(3) + 7(2)) = 4(6 + 14) = 4(20) = 80 ✓
Answer: 8 terms are needed.
Section 10.6

Geometric Sequences

A geometric sequence is a sequence in which each term is obtained from the previous term by multiplying by a fixed constant called the common ratio r. If you divide any term by the previous term and always get the same result, the sequence is geometric: r = aₙ / aₙ₋₁. Just as arithmetic sequences correspond to linear functions, geometric sequences correspond to exponential functions.

SequenceCommon RatioBehavior
2, 6, 18, 54, 162, …r = 3Exponential growth
1000, 500, 250, 125, …r = ½Exponential decay
1, −2, 4, −8, 16, …r = −2Alternating growth
Geometric Sequence — Explicit Formula aₙ = a₁ · r^(n−1)
Example 10.6.1
Finding a Term of a Geometric Sequence
  • ProblemA geometric sequence has a₁ = 5 and r = 3. Find a₇.
  • Solvea₇ = 5 · 3^(7−1) = 5 · 3⁶ = 5 · 729 = 3645
Answer: a₇ = 3645
Example 10.6.2
Finding the Common Ratio and Explicit Formula
  • ProblemA geometric sequence has a₃ = 20 and a₆ = 160. Find the explicit formula.
  • ThinkThe ratio of two terms eliminates a₁. Form a₆/a₃ = r³, then solve for r, then find a₁.
  • Solve a₆/a₃ = (a₁r⁵)/(a₁r²) = r³ = 160/20 = 8 → r = 2 From a₃ = a₁ · r² = 20: a₁ · 4 = 20 → a₁ = 5 aₙ = 5 · 2^(n−1)
  • Checka₃ = 5 · 4 = 20 ✓ a₆ = 5 · 32 = 160 ✓
Answer: aₙ = 5 · 2^(n−1)
Section 10.7

Geometric Series

A geometric series is the sum of the terms of a geometric sequence. To derive the finite sum formula, multiply both sides of the sum by r and subtract the original expression. All intermediate terms cancel, leaving only the first and last.

Finite Geometric Series — Sum Formula (r ≠ 1) Sₙ = a₁(1 − rⁿ) / (1 − r)
Example 10.7.1
Finding a Geometric Series Sum
  • ProblemFind the sum of the first 8 terms of the geometric sequence 3, 6, 12, 24, …
  • Representa₁ = 3, r = 2, n = 8
  • Solve S₈ = 3(1 − 2⁸) / (1 − 2) = 3(1 − 256) / (−1) = 3(−255) / (−1) = 765
  • CheckTerms: 3, 6, 12, 24, 48, 96, 192, 384. Adding: 3+6=9, +12=21, +24=45, +48=93, +96=189, +192=381, +384=765 ✓
Answer: S₈ = 765
Example 10.7.2
Geometric Series in Finance — Annuity Value
  • ProblemYou invest $500 at the start of each year for 5 years in an account paying 6% annual interest. How much is in the account at the end of 5 years?
  • ThinkDeposit 1 grows for 5 years: 500(1.06)⁵. Deposit 2 for 4 years: 500(1.06)⁴. And so on down to deposit 5 growing for 1 year. The total is a geometric series.
  • Solve FV = 500(1.06)¹ + 500(1.06)² + 500(1.06)³ + 500(1.06)⁴ + 500(1.06)⁵ a₁ = 500(1.06) = 530, r = 1.06, n = 5 FV = 530(1 − 1.06⁵) / (1 − 1.06) = 530(1 − 1.33823) / (−0.06) ≈ 530(0.33823) / 0.06 ≈ $2,987.67
Answer: Approximately $2,987.67
Section 10.8

Infinite Geometric Series

It seems impossible for an infinite sum to be finite — you are adding infinitely many numbers. But consider ½ + ¼ + ⅛ + ⅟₁₆ + ⋯. The partial sums are ½, ¾, ⅞, ¹⁵⁄₁₆, … — always less than 1 and approaching 1. As n → ∞, Sₙ → 1. The infinite series converges to a finite value when the terms shrink quickly enough, specifically when |r| < 1.

Infinite Geometric Series — Convergence

When |r| < 1, the series converges: S = a₁ / (1 − r)

When |r| ≥ 1, the series diverges — no finite sum exists.

The derivation: in the finite formula Sₙ = a₁(1 − rⁿ)/(1 − r), when |r| < 1, rⁿ → 0 as n → ∞, leaving S = a₁/(1 − r).

Example 10.8.1
Sum of an Infinite Geometric Series
  • ProblemFind the sum of the infinite series 8 + 4 + 2 + 1 + ½ + ⋯
  • Representa₁ = 8, r = ½. Since |r| = ½ < 1, the series converges.
  • SolveS = 8 / (1 − ½) = 8 / (½) = 16
  • CheckPartial sums: 8, 12, 14, 15, 15.5, 15.75, … — approaching 16 ✓
Answer: S = 16
Example 10.8.2
Converting a Repeating Decimal to a Fraction
  • ProblemExpress 0.363636… as a fraction using the infinite series formula.
  • Represent0.363636… = 36/100 + 36/10000 + 36/1000000 + ⋯ This is an infinite geometric series with a₁ = 36/100 and r = 1/100.
  • Solve S = (36/100) / (1 − 1/100) = (36/100) / (99/100) = 36/99 = 4/11
  • Check4/11 = 0.363636… ✓
Answer: 0.3̄6̄ = 4/11
Conceptual Note — Every Repeating Decimal Is Rational

This example shows that every repeating decimal equals a rational number. The infinite geometric series formula provides the proof. This is also your first serious encounter with a limit: the partial sums form a sequence that approaches S as n → ∞ — exactly the concept studied rigorously in Calculus I and II.

Sections 10.9 – 10.11
Sigma Notation, Partial Sums, and Conjectures
Sigma Notation · Properties · Partial Sums · Convergence · Patterns and Conjectures
Section 10.9

Sigma Notation

Writing out every term of a series is tedious when the number of terms is large. The Greek capital letter Σ (sigma) means "sum." A sigma expression has three components: the index of summation (a dummy variable tracking position), the lower limit (starting index), and the upper limit (ending index). The general term specifies what is being summed.

Sigma Notation Structure

Σ(k=m to n) aₖ means: evaluate aₖ for every integer from k = m to k = n, then add all values.

The index k is a dummy variable — Σ(k=1 to n) k² and Σ(j=1 to n) j² are identical expressions. The letter used does not matter.

Example 10.9.1
Evaluating a Sigma Expression
  • ProblemEvaluate Σ(k=1 to 5) (2k + 1).
  • Solve k=1: 2(1)+1 = 3 k=2: 2(2)+1 = 5 k=3: 2(3)+1 = 7 k=4: 2(4)+1 = 9 k=5: 2(5)+1 = 11 Sum: 3+5+7+9+11 = 35
Answer: 35
Example 10.9.2
Using Properties to Evaluate a Large Sum
  • ProblemEvaluate Σ(k=1 to 100) (3k − 2).
  • ThinkSplit using the sum rule, then apply the standard formula Σk = n(n+1)/2.
  • Solve Σ(3k − 2) = 3·Σk − Σ2 = 3 · (100·101/2) − 100·2 = 3(5050) − 200 = 15150 − 200 = 14950
Answer: 14,950

Standard Summation Formulas

FormulaStatement
Σ c = n · cSum of a constant over n terms
Σ k = n(n+1)/2Sum of first n positive integers
Σ k² = n(n+1)(2n+1)/6Sum of first n perfect squares
Σ k³ = [n(n+1)/2]²Sum of first n perfect cubes

These formulas are proved using mathematical induction in Section 10.12.

Section 10.10

Partial Sums

The n-th partial sum of a sequence is the sum of the first n terms: Sₙ = a₁ + a₂ + ⋯ + aₙ. The partial sums themselves form a new sequence S₁, S₂, S₃, … that records how the running total grows. For infinite series, convergence is defined by the behavior of this sequence of partial sums: if Sₙ approaches a finite limit L as n → ∞, the series converges to L. If the partial sums grow without bound or oscillate, the series diverges.

Example 10.10.1
Generating and Interpreting Partial Sums
  • ProblemFor aₙ = (1/3)ⁿ, list the first six partial sums and describe convergence behavior.
  • Solve a₁ = 1/3 ≈ 0.333 S₁ ≈ 0.333 S₂ = 1/3 + 1/9 = 4/9 ≈ 0.444 S₃ = 4/9 + 1/27 = 13/27 ≈ 0.481 S₄ ≈ 0.494 S₅ ≈ 0.498 S₆ ≈ 0.499 Infinite sum: a₁=1/3, r=1/3 → S = (1/3)/(1−1/3) = (1/3)/(2/3) = 1/2
  • CheckPartial sums are increasing and approaching 1/2 from below — convergence confirmed.
Graph instruction: Plot (1, S₁) through (6, S₆). Draw a dashed line at y = 1/2. Label it "Infinite Sum = 1/2." Points approach the line from below and never exceed it.
Section 10.11

Patterns and Conjectures

Mathematics progresses by recognizing patterns and asking whether those patterns always hold. A conjecture is a statement that appears true based on observed examples but has not yet been proven. Forming a good conjecture is a valuable skill — but it is only the first step.

Example 10.11.1
Forming a Conjecture from a Pattern
  • ProblemObserve: 1 = 1; 1+3 = 4; 1+3+5 = 9; 1+3+5+7 = 16; 1+3+5+7+9 = 25. Identify the pattern and state a conjecture.
  • ThinkThe sums 1, 4, 9, 16, 25 are perfect squares. The sum of the first n odd positive integers appears to equal n².
  • ConjectureFor every positive integer n: Σ(k=1 to n) (2k − 1) = n²
Caution: Verifying five cases does not prove the formula for all n. A proof by mathematical induction is required — see Section 10.12.
Why Observation Is Not Proof

The expression n² − n + 41 produces prime numbers for n = 1, 2, 3, …, 40 — forty consecutive cases. But when n = 41: 41² − 41 + 41 = 41² = 1681 = 41 × 41, which is not prime. The pattern broke at case 41. This is why mathematicians do not accept a conjecture as true based on examples alone, no matter how many cases have been verified.

Section 10.12
Mathematical Induction
Base Case · Inductive Hypothesis · Inductive Step · Three Complete Proofs
Section 10.12

Mathematical Induction

Mathematical induction is a proof technique that establishes a statement P(n) for all positive integers n. It works by proving two things: (1) P(1) is true (the base case), and (2) whenever P(k) is true for some positive integer k, then P(k + 1) must also be true (the inductive step). The logic is like a chain of falling dominoes: if the first falls (base case), and whenever any domino falls the next falls (inductive step), then all dominoes fall.

The inductive hypothesis is an assumption used within the proof. You assume P(k) is true for a specific k, then derive P(k+1) from it using algebra. You are not assuming what you want to prove for all n — you are assuming it for a specific k and showing the logical consequence for k + 1.

Structure of a Mathematical Induction Proof

Step 1 — State: Write out clearly what statement P(n) claims.

Step 2 — Base Case: Verify that P(1) is true.

Step 3 — Inductive Hypothesis: Assume P(k) is true for some positive integer k.

Step 4 — Inductive Step: Using the assumption that P(k) is true, prove that P(k+1) is also true.

Step 5 — Conclude: By the principle of mathematical induction, P(n) is true for all positive integers n.

Proof by Mathematical Induction — Example 10.12.1
Claim: For all positive integers n, Σ(k=1 to n) k = n(n+1)/2
Base Case (n=1)Left side: Σ(k=1 to 1) k = 1. Right side: 1(1+1)/2 = 2/2 = 1. Both sides equal 1. ✓
Inductive HypothesisAssume the formula holds for some positive integer k: Σ(j=1 to k) j = k(k+1)/2.
Inductive StepMust show the formula holds for k+1, i.e., Σ(j=1 to k+1) j = (k+1)(k+2)/2.

Starting from the left side:
Σ(j=1 to k+1) j = [Σ(j=1 to k) j] + (k+1)
= k(k+1)/2 + (k+1) [by the inductive hypothesis]
= (k+1)[k/2 + 1]
= (k+1) · (k+2)/2
= (k+1)(k+2)/2 ✓
ConclusionBy the principle of mathematical induction, Σ(k=1 to n) k = n(n+1)/2 for all positive integers n.
■
Proof by Mathematical Induction — Example 10.12.2
Claim: For all positive integers n, Σ(k=1 to n) (2k−1) = n²
Base Case (n=1)Left: 2(1)−1 = 1. Right: 1² = 1. ✓
Inductive HypothesisAssume Σ(j=1 to k) (2j−1) = k² for some positive integer k.
Inductive StepΣ(j=1 to k+1)(2j−1) = [Σ(j=1 to k)(2j−1)] + (2(k+1)−1)
= k² + 2k + 2 − 1
= k² + 2k + 1
= (k+1)² ✓
ConclusionBy mathematical induction, Σ(k=1 to n)(2k−1) = n² for all positive integers n.
■
Proof by Mathematical Induction — Example 10.12.3
Claim: For all non-negative integers n and r ≠ 1, Σ(k=0 to n) rᵏ = (1 − r^(n+1)) / (1 − r)
Base Case (n=0)Left: r⁰ = 1. Right: (1−r¹)/(1−r) = (1−r)/(1−r) = 1. ✓
Inductive HypothesisAssume Σ(j=0 to k) rʲ = (1 − r^(k+1))/(1 − r) for some non-negative integer k.
Inductive StepΣ(j=0 to k+1) rʲ = [Σ(j=0 to k) rʲ] + r^(k+1)
= (1 − r^(k+1))/(1 − r) + r^(k+1)
= [1 − r^(k+1) + r^(k+1)(1 − r)] / (1 − r)
= [1 − r^(k+1) + r^(k+1) − r^(k+2)] / (1 − r)
= (1 − r^(k+2)) / (1 − r) ✓

This matches the formula for n = k+1.
ConclusionBy mathematical induction, the geometric sum formula holds for all non-negative integers n.
■
Student Warning — The Inductive Step Must Go Forward

The inductive step must show that assuming P(k) logically forces P(k+1). Many students accidentally try to assume P(k+1) and derive P(k) — the wrong direction. Others treat the inductive hypothesis as something to prove within the inductive step, creating circular reasoning. Always start from the left side of the (k+1) case, apply the inductive hypothesis partway through, and arrive at the right side.

When to Use Induction

Mathematical induction is the right tool when you have a formula or statement involving n that you believe is true for all positive integers and you can show algebraically that P(k) forces P(k+1). Induction is not a tool for discovering formulas — it is a tool for proving formulas already conjectured through pattern recognition.

Sections 10.13 – 10.14
Applications and Common Mistakes
Finance · Computing · Science · Common Errors
Section 10.13

Sequences in Finance, Computing, and Science

Finance — Compound Interest and Annuities

Geometric sequences and series are the mathematical foundation of personal finance. If a principal P earns interest at rate r per period, the balance after n periods is Aₙ = P(1 + r)ⁿ — a geometric sequence with ratio (1 + r). An ordinary annuity (deposits at end of each period) with payment C and interest rate r over n periods has future value FV = C · [(1 + r)ⁿ − 1] / r, derived directly from the geometric series formula.

Example 10.13.1
Retirement Savings — An Annuity Calculation
  • ProblemA 25-year-old deposits $300 per month into a retirement account earning 7% annual interest, compounded monthly. How much is in the account at age 65?
  • RepresentMonthly rate r = 0.07/12 ≈ 0.005833. Number of periods n = 40 × 12 = 480 months. C = $300.
  • Solve FV = 300 · [(1.005833)^480 − 1] / 0.005833 ≈ 300 · [14.974 − 1] / 0.005833 ≈ 300 · 13.974 / 0.005833 ≈ $718,740
  • InterpretTotal deposited: 300 × 480 = $144,000. The remaining $574,740 comes from compound interest — a striking illustration of geometric growth over time.
Answer: Approximately $718,740 at age 65.

Computing — Recursive Algorithms and Complexity

Many computing algorithms are defined recursively. Binary search halves a sorted list at each step; the maximum number of steps is approximately log₂ n. Merge sort splits a list, sorts each half, and merges; the recurrence T(n) = 2T(n/2) + n has solution T(n) = n log₂ n, making it far more efficient than simpler O(n²) sorting algorithms for large n. The formula Σ(k=1 to n) k = n(n+1)/2 appears constantly in complexity analysis — a nested loop where the inner loop runs k times during outer iteration k performs approximately n²/2 total operations.

Science — Radioactive Decay

Radioactive decay follows a geometric sequence. If a substance has half-life h, the amount remaining after n half-life periods is Aₙ = A₀ · (1/2)ⁿ.

Example 10.13.2
Radioactive Decay — Carbon-14
  • ProblemCarbon-14 has a half-life of approximately 5730 years. A sample initially contains 80 grams. How much remains after 5 half-lives?
  • SolveA₅ = 80 · (1/2)⁵ = 80 · (1/32) = 2.5 grams
  • InterpretAfter 5 × 5730 = 28,650 years, only 2.5 grams of the original 80-gram sample remains.
Answer: 2.5 grams remain after 5 half-lives.
Section 10.14

Common Sequence and Series Mistakes

Mistake 1 — Off-by-One Error in the Exponent
aₙ = 3 · 2ⁿ (gives a₁ = 6, not 3)

Error: The exponent should be n − 1, not n. At n = 1 the formula must give the first term, requiring 2⁰ = 1.

Correct: aₙ = 3 · 2^(n−1) → a₁ = 3 · 2⁰ = 3 ✓
Mistake 2 — Confusing Arithmetic and Geometric Sequences
Applying Sₙ = n/2(a₁ + aₙ) to a geometric series

Error: The arithmetic sum formula applies only when a constant is added at each step. If a constant is multiplied, use the geometric formula.

Correct: For geometric series, use Sₙ = a₁(1 − rⁿ)/(1 − r).
Mistake 3 — Using the Infinite Sum Formula When |r| ≥ 1
S = 6/(1−1.5) = 6/(−0.5) = −12 for the series 6 + 9 + 13.5 + ⋯

Error: The ratio r = 1.5 and |r| = 1.5 ≥ 1. The series diverges — no finite sum exists.

Correct: First check |r| < 1. Since |1.5| ≥ 1, write "the series diverges."
Mistake 4 — Sigma Index Confusion
Σ(k=0 to n) treated as having n terms

Error: When the index starts at 0, the sum Σ(k=0 to n) has n + 1 terms (k = 0, 1, 2, …, n), not n terms.

Correct: Σ(k=0 to n) has n+1 terms. Σ(k=1 to n) has n terms.
Mistake 5 — Circular Reasoning in the Inductive Step
"Assume P(k+1), then derive P(k) — done!"

Error: The inductive step must go forward: assume P(k), then derive P(k+1). Going from P(k+1) backward to P(k) proves nothing.

Correct: Start from the left side of the P(k+1) statement, apply the inductive hypothesis midway, and arrive at the right side of P(k+1).
Mistake 6 — Omitting the Base Case in Induction
"I proved the inductive step, so I'm done."

Error: Both the base case and the inductive step are required. The inductive step alone only says "if the chain starts, it continues." Without the base case, the chain never starts.

Correct: Always prove the base case first, then the inductive step.
Chapter 10 — Back Matter

Chapter Summary

Key Terms Review

sequence
A function with domain equal to the set of positive integers; outputs are called terms.
explicit formula
A formula that gives aₙ directly as a function of n without requiring prior terms.
recursive formula
A formula that defines aₙ in terms of previous terms, together with one or more initial conditions.
arithmetic sequence
A sequence with constant common difference d; explicit formula aₙ = a₁ + (n−1)d.
common difference
The constant d added at each step in an arithmetic sequence: d = aₙ − aₙ₋₁.
geometric sequence
A sequence with constant common ratio r; explicit formula aₙ = a₁ · r^(n−1).
common ratio
The constant r multiplied at each step in a geometric sequence: r = aₙ / aₙ₋₁.
series
The sum of the terms of a sequence: S = a₁ + a₂ + a₃ + ⋯
partial sum
The sum of the first n terms of a sequence: Sₙ = a₁ + a₂ + ⋯ + aₙ.
sigma notation
Compact summation notation using the Greek letter Σ with index, lower limit, upper limit, and general term.
index of summation
The dummy variable in sigma notation that tracks position; the letter used does not affect the sum's value.
convergence
The property of an infinite series whose partial sums approach a finite limit.
divergence
The property of an infinite series whose partial sums grow without bound or oscillate without settling.
mathematical induction
A proof technique that establishes P(n) for all positive integers by proving a base case and an inductive step.
base case
The initial case verified in a proof by induction, typically P(1).
inductive hypothesis
The assumption that P(k) is true for some positive integer k, used within the inductive step.
inductive step
The logical argument showing that if P(k) is true, then P(k+1) must also be true.
conjecture
A statement believed to be true based on pattern observation but not yet proven.
annuity
A financial instrument involving a series of equal periodic payments; its future value is computed using the geometric series formula.
Fibonacci sequence
The recursive sequence F₁=1, F₂=1, Fₙ=Fₙ₋₁+Fₙ₋₂; each term is the sum of the two preceding terms.
geometric sum formula
The formula Sₙ = a₁(1 − rⁿ)/(1 − r) for the sum of n terms of a geometric sequence with r ≠ 1.

Concept Review Questions

  1. Explain the difference between a sequence and a series.
  2. What is the difference between an explicit formula and a recursive formula? Give an example of each.
  3. How can you tell whether a sequence is arithmetic? Geometric? What is the key distinction?
  4. Write the arithmetic sequence formula aₙ = a₁ + (n−1)d and explain what each variable represents.
  5. Write the geometric sequence formula aₙ = a₁ · r^(n−1) and explain why the exponent is n − 1, not n.
  6. State the arithmetic series sum formula and explain the idea behind Gauss's pairing argument.
  7. State the geometric series sum formula and explain its derivation in your own words.
  8. What condition must r satisfy for an infinite geometric series to converge? What happens when this condition fails?
  9. What is a partial sum? How do partial sums relate to convergence?
  10. Explain sigma notation. What does each part of Σ(k=2 to 7)(k² − 1) represent?
  11. What are the two required steps in a proof by mathematical induction? Why are both necessary?
  12. What is the inductive hypothesis? How is it used in the inductive step?
  13. A student checks a formula for n = 1 through n = 50 and it works every time. Does that prove the formula for all n? Explain.
  14. Describe one application of geometric sequences outside of pure mathematics.

Skill Practice

Part AExplicit Formulas
  1. aₙ = 4n − 3: Find a₁, a₅, a₂₀, a₁₀₀.
  2. aₙ = (−1)ⁿ · n: Find the first six terms.
  3. aₙ = n/(n+1): Find first five terms. What value does it approach?
  4. aₙ = 3·(2/3)^(n−1): Find a₁, a₄, a₁₀.
  5. Write a formula for: 4, 7, 10, 13, 16, …
  6. Write a formula for: 1, −1, 1, −1, 1, …
  7. Write a formula for: ½, ¼, ⅛, ¹⁄₁₆, …
  8. Write a formula for: 2, 6, 12, 20, 30, … (Hint: n(n+1))
Part BArithmetic Sequences and Series
  1. Is 5, 11, 17, 23, 29 arithmetic? State d.
  2. a₁ = −4, d = 6. Find a₁₅.
  3. a₇ = 31, a₁₅ = 63. Find a₁, d, and aₙ.
  4. Insert three arithmetic means between 10 and 26.
  5. Find S₅₀ for 1 + 2 + 3 + ⋯ + 50.
  6. Find the sum 5 + 9 + 13 + ⋯ + 97.
  7. a₁ = 7, d = −2. Find S₃₀.
  8. A theater has 20 rows; row 1 has 15 seats; each row has 2 more than the row before. Total seats?
Part CGeometric Sequences and Series
  1. Is 2, 6, 18, 54 geometric? State r.
  2. Is 3, −6, 12, −24 geometric? State r.
  3. a₁ = 4, r = 3. Find a₆.
  4. a₂ = 10, a₅ = 80. Find r, a₁, and aₙ.
  5. Find S₆ for 2, 6, 18, …
  6. Find S₈ for 4, −8, 16, −32, …
  7. Does Σ(n=1 to ∞) 5·(¼)^(n−1) converge? Find the sum.
  8. Does Σ(n=1 to ∞) 3·(1.02)^(n−1) converge? Explain.
Part DRepeating Decimals and Infinite Series
  1. Express 0.7̄ as a fraction.
  2. Express 0.1̄2̄3̄ as a fraction.
  3. A ball drops 12 ft; each bounce is ⅔ the last. Total distance?
  4. A pendulum swings 20 cm first arc; each subsequent arc is 90%. Total distance?
  5. Partial sums: list S₁ through S₆ for aₙ = (½)ⁿ. What sum do they approach?
  6. Evaluate Σ(k=1 to 5) (k² − 1).
  7. Evaluate Σ(k=0 to 6) 3ᵏ.
  8. Evaluate Σ(k=1 to 100) (5k − 3).

Mathematical Induction Practice

Prove each statement using mathematical induction. Show all three steps: base case, inductive hypothesis, and inductive step.

  1. Prove: Σ(k=1 to n) k = n(n+1)/2
  2. Prove: Σ(k=1 to n) k² = n(n+1)(2n+1)/6
  3. Prove: Σ(k=1 to n) 2^(k−1) = 2ⁿ − 1
  4. Prove: Σ(k=1 to n) k(k+1) = n(n+1)(n+2)/3
  5. Prove: 4ⁿ − 1 is divisible by 3 for all positive integers n.
  6. Prove: n! ≥ 2^(n−1) for all positive integers n ≥ 1.
  7. Prove: 2n + 1 ≤ 2ⁿ for all integers n ≥ 3.
  8. (Challenge) Prove the geometric sum formula Sₙ = a₁(1 − rⁿ)/(1 − r) using induction on n.

Word Problems and Applications

Define all variables, set up the appropriate sequence or series, evaluate, and interpret your answer in context.

  1. Loan balance: A $10,000 loan charges 1% interest per month. With no payments, the balance after n months is 10000·(1.01)ⁿ. What is the balance after 2 years?
  2. Drug concentration: A patient takes 200 mg every 6 hours; the body eliminates 30% every 6 hours. Write a recursive formula for the drug amount just before the (n+1)-th dose. What does the amount approach long-term?
  3. Architecture: A spiral of squares — each square has side length equal to half the previous; first square has side 8 cm. What is the total area of all infinitely many squares?
  4. Investment comparison: Plan A deposits $1,000/year for 10 years at 5% annually. Plan B makes a single $10,000 deposit at year 0. Compare balances at the end of 10 years.
  5. Algorithm analysis: An algorithm performs k operations in step k for k = 1, 2, …, n. Write a formula for total operations after n steps. For n = 100, what is the total?
  6. Zeno's paradox: Zeno argued motion is impossible: to travel any distance, first travel half, then half the remainder, ad infinitum. Use the infinite geometric series to resolve this paradox mathematically. What does the series tell us about the total distance?

Error Analysis

Each problem shows a student's work containing an error. Identify the error and write the correct solution.

Problem A
"The sequence 5, 15, 45, 135 is arithmetic with common difference d = 10."

What is the error? What type of sequence is this? State the correct classification and the key quantity.

Problem B
"Sum of 6 + 9 + 13.5 + … = 6/(1 − 1.5) = 6/(−0.5) = −12"

What is the error? What is the correct conclusion about this series?

Problem C — Induction Error
"Inductive step: Assume Σ(k=1 to n) k = n(n+1)/2. We want to show this, and we have shown it, so the proof is complete."

Identify the logical error. What has been assumed versus proven? Rewrite the inductive step correctly.

Problem D — Off-by-One Exponent
For a geometric sequence with a₁ = 4: student writes aₙ = 4 · 3ⁿ and claims a₁ = 4.

Check: what does 4 · 3¹ give? What is the correct formula? Fix it.

Problem E — Missing Base Case
"I proved P(k) → P(k+1) for all k. My induction proof is complete."

What essential step is missing? Why does omitting it make the proof invalid? Give an analogy to explain.

Chapter 10 Checkpoint

Chapter 10 Checkpoint
Complete without assistance — assesses all key skills and concepts from Chapter 10
Section 1 — Vocabulary and Concepts
  1. Explain what it means to say a sequence is a function. What is its domain? How does its graph differ from a continuous function's graph?
  2. What distinguishes an arithmetic sequence from a geometric sequence? Give one example of each.
  3. What condition must the common ratio r satisfy for an infinite geometric series to converge?
Section 2 — Sequences
  1. Let aₙ = 5n − 7. Find a₁, a₄, and a₃₀. Is this sequence arithmetic, geometric, or neither?
  2. An arithmetic sequence has a₁ = 12 and d = −4. Find a₂₅.
  3. A geometric sequence has a₁ = 81 and r = ⅓. Write the first five terms.
  4. A sequence is defined by a₁ = 2 and aₙ = 3aₙ₋₁ − 1. Find the first five terms.
Section 3 — Series and Sigma Notation
  1. An arithmetic sequence has a₁ = 12 and d = −4. Find S₂₅.
  2. A geometric sequence has a₁ = 81 and r = ⅓. Find S₅.
  3. Evaluate: Σ(k=1 to 6) 4·(½)^(k−1)
  4. Does Σ(n=0 to ∞) (¾)ⁿ converge? If so, find the sum.
Section 4 — Mathematical Induction
  1. Prove by induction: Σ(k=1 to n) 2k = n(n+1). Show all steps clearly.
  2. A student wants to prove that 2ⁿ ≥ n + 1 for all positive integers n. Verify the base case n = 1. State the inductive hypothesis. Carry out the inductive step.
Section 5 — Applications
  1. You deposit $1,000 at the beginning of each year for 10 years into an account earning 4% annual interest compounded annually. Set up the geometric series for the total future value and evaluate it.
  2. A ball is dropped from 16 feet and bounces back to ¾ of its previous height on each bounce. What is the total vertical distance traveled?
Section 6 — Error Analysis
  1. A student claims the sum of the infinite series 3 + 4.5 + 6.75 + ⋯ is 3/(1 − 1.5) = −6. Identify all errors. What is the correct conclusion?
  2. A student writes the geometric sequence with a₁ = 2 and r = 5 as aₙ = 2·5ⁿ. Find and correct the error. What does a₁ equal under the student's formula?
Looking Ahead
Bridge to Calculus and Infinite Analysis

Chapter 10 is a turning point in the MGU Mathematics Series. This is the chapter where finite, step-by-step mathematics first reaches toward the infinite — and where you discover that an infinite process can produce a precise finite result. The infinite geometric series formula S = a₁/(1 − r) is your first serious encounter with a limit. The partial sums S₁, S₂, S₃, … form a sequence that approaches S as n → ∞, which is exactly the concept of a limit of a sequence studied rigorously in Calculus I and II.

In Calculus I, you will meet limits of functions — the continuous analog of sequence limits — along with derivatives defined as limits of difference quotients. In Calculus II, infinite series become a central topic, and you will apply convergence tests (ratio test, integral test, comparison test) to determine whether series converge. Taylor series and power series express sin(x), cos(x), and eˣ as infinite polynomial sums — among the most powerful tools in all of mathematics and physics.

In Discrete Mathematics and Computer Science, recurrences, generating functions, and strong induction extend the ideas of this chapter to algorithm analysis, data structures, and combinatorics. Chapter 11 continues Part IV by turning to analytic geometry — conic sections, where algebra and geometry meet in a different way, through the shapes produced by quadratic equations in two variables.

MGU Library Connection
  • Sequence and Series Formula Sheet — Appendix L of this textbook
  • Algebra II, Chapter 11: Sequences, Series, and Probability (prerequisite review)
  • MGU Calculus I, Chapters 1–2: Limits of Functions and Sequences
  • MGU Calculus II, Chapter 8: Infinite Series and Convergence Tests
  • MGU Calculus II, Chapter 10: Taylor and Maclaurin Series
  • MGU Discrete Mathematics, Chapter 5: Mathematical Induction and Recursion
  • MGU Dictionary entries: sequence, series, partial sum, convergence, induction, sigma notation