The Central Idea
A sequence is an ordered list of terms.
For example,
\[ 2,\ 4,\ 6,\ 8,\ 10,\ldots \]is a sequence.
The numbers themselves are called terms, and each term has a particular position in the sequence.
We can label those positions with indices:
\[ a_1=2,\qquad a_2=4,\qquad a_3=6,\qquad a_4=8,\qquad a_5=10. \]The subscript tells us which term we are discussing.
A Sequence Is Ordered
Order is essential to a sequence.
Consider
\[ 3,\ 5,\ 7 \]and
\[ 7,\ 5,\ 3. \]These sequences contain the same three numbers, but they are different sequences because the terms occur in different positions.
In the first sequence,
\[ a_1=3, \]while in the second,
\[ a_1=7. \]A term's identity within a sequence therefore includes its position.
Sequences and Sets Are Different
A sequence should not be confused with a set.
A set records membership. A sequence records both membership and order.
For example, the sequences
\[ 1,\ 2,\ 3 \]and
\[ 3,\ 2,\ 1 \]have different orders.
By contrast, the sets
\[ \{1,2,3\} \]and
\[ \{3,2,1\} \]represent the same set.
This distinction becomes especially important when repeated values appear.
Repeated Terms Are Allowed
A sequence may contain the same value more than once.
For example,
\[ 1,\ 4,\ 4,\ 7,\ 4,\ldots \]is a perfectly valid sequence.
Here,
\[ a_2=4,\qquad a_3=4,\qquad a_5=4. \]The value \(4\) appears at several different indices.
This is another reason that the index is important: identical values can occupy different positions.
What Is a Term?
A term is one individual member of a sequence at a specified position.
In the sequence
\[ 5,\ 10,\ 15,\ 20,\ 25,\ldots \]the first term is
\[ 5, \]the second term is
\[ 10, \]and the fifth term is
\[ 25. \]Using sequence notation,
\[ a_1=5,\qquad a_2=10,\qquad a_5=25. \]What Is an Index?
An index is a label that identifies the position of a term.
In
\[ a_7, \]the \(7\) is the index.
The notation \(a_7\) means
the term of sequence \(a\) associated with index \(7\).
If
\[ a_7=31, \]then the term at index \(7\) has the value \(31\).
The Subscript Is Not Multiplication
Sequence notation uses a subscript, not ordinary multiplication.
The expression
\[ a_3 \]does not normally mean
\[ a\cdot3. \]It means the term of sequence \(a\) whose index is \(3\).
For example, if
\[ a_1=4,\qquad a_2=9,\qquad a_3=14, \]then
\[ a_3=14. \]The General Term
Instead of naming one particular position, we often use a variable index.
The notation
\[ a_n \]means the term of sequence \(a\) at index \(n\).
It is commonly called the general term or the \(n\)th term.
The symbol \(n\) usually represents an integer from the indexing set of the sequence.
The Meaning of the \(n\)th Term
Suppose a sequence is defined by
\[ a_n=3n+1. \]The formula tells us how the value of a term depends on its index.
For \(n=1\),
\[ a_1=3(1)+1=4. \]For \(n=2\),
\[ a_2=3(2)+1=7. \]For \(n=3\),
\[ a_3=3(3)+1=10. \]Thus the sequence begins
\[ 4,\ 7,\ 10,\ldots \]Sequence Names
The letter used in sequence notation identifies the sequence.
We might have
\[ a_n,\qquad b_n,\qquad c_n. \]These can represent three different sequences.
For example,
\[ a_n=n, \] \[ b_n=n^2, \]and
\[ c_n=2^n. \]At \(n=3\),
\[ a_3=3,\qquad b_3=9,\qquad c_3=8. \]Sequences Usually Begin at \(1\)
In many elementary examples, indexing begins with
\[ n=1. \]Then the sequence is written
\[ a_1,\ a_2,\ a_3,\ a_4,\ldots \]and \(a_1\) is called the first term.
This convention is common, but it is not universal.
Sequences Can Begin at \(0\)
A sequence may instead begin with index \(0\):
\[ a_0,\ a_1,\ a_2,\ a_3,\ldots \]This convention is especially common in computer science, combinatorics, recurrence relations, power series, and other areas of mathematics.
If
\[ a_n=2^n \]for \(n\geq0\), then
\[ a_0=1,\qquad a_1=2,\qquad a_2=4,\qquad a_3=8. \]The resulting sequence is
\[ 1,\ 2,\ 4,\ 8,\ldots \]The Starting Index Must Be Known
The formula alone does not always tell us where a sequence begins.
Suppose
\[ a_n=n^2. \]If the indexing begins at \(n=1\), the sequence starts
\[ 1,\ 4,\ 9,\ 16,\ldots \]If the indexing begins at \(n=0\), it starts
\[ 0,\ 1,\ 4,\ 9,\ 16,\ldots \]The rule is the same, but the indexed sequence is different.
Sequences Can Begin at Other Indices
There is no requirement that a sequence begin at \(0\) or \(1\).
For example, we may define
\[ a_n=n^2 \qquad\text{for }n\geq5. \]Then the first indexed term in the specified sequence is
\[ a_5=25. \]The sequence continues
\[ 25,\ 36,\ 49,\ 64,\ldots \]In advanced work, specifying the indexing set precisely prevents ambiguity.
The Indexing Set
The collection of allowed indices is called the indexing set or index set.
A typical infinite sequence may use
\[ \{1,2,3,\ldots\}. \]Another may use
\[ \{0,1,2,3,\ldots\}. \]A finite sequence might use
\[ \{1,2,3,4,5\}. \]The indexing set tells us which terms actually belong to the sequence.
A Sequence as a Function
A sequence can be viewed as a special kind of function whose inputs are discrete indices.
For example,
\[ a_n=2n+3 \]can be understood as assigning an output to each allowed integer \(n\):
\[ n\longmapsto a_n. \]If the sequence begins at \(n=1\), then
\[ 1\mapsto5,\qquad 2\mapsto7,\qquad 3\mapsto9. \]This function viewpoint will become useful when we study discrete graphs, recurrence relations, iteration, and convergence.
Discrete Inputs
Ordinary functions may be defined on continuous intervals such as
\[ 0\leq x\leq10. \]A standard sequence instead uses separated input values such as
\[ 1,\ 2,\ 3,\ 4,\ldots \]There is no term at index
\[ 2.5 \]unless the sequence has explicitly been defined with such an indexing system.
Standard numerical sequences therefore provide an important example of discrete mathematics.
Listing a Sequence
One way to describe a sequence is simply to list its terms.
For example,
\[ 2,\ 5,\ 8,\ 11,\ 14,\ldots \]communicates several initial values.
The ellipsis
\[ \ldots \]suggests that the sequence continues.
However, listing terms alone does not always provide a complete mathematical definition.
An Ellipsis Does Not Prove the Pattern
Consider
\[ 1,\ 2,\ 4,\ 8,\ldots \]A natural guess is that each term doubles, giving
\[ 16,\ 32,\ldots \]But a finite list of observed values can be continued in many different ways unless a rule is specified.
Patterns help us form conjectures, but a precise sequence definition tells us which continuation is intended.
Writing a Sequence with Indexed Terms
Instead of listing only values, we can display both indices and terms:
\[ a_1=4,\quad a_2=7,\quad a_3=10,\quad a_4=13,\ldots \]This notation makes the correspondence between position and value explicit.
We can also write
\[ (a_1,a_2,a_3,a_4,\ldots). \]Parentheses are often useful because they emphasize ordered structure.
Sequence Notation with Braces
Sequences are sometimes written using notation such as
\[ \{a_n\}. \]Depending on the textbook or mathematical context, this notation may mean the sequence of terms \(a_n\).
However, braces are also standard notation for sets.
Because sequences preserve order and repeated terms while sets do not, context matters.
When there is any risk of ambiguity, notation such as
\[ (a_n)_{n=1}^{\infty} \]makes the indexed nature of the sequence explicit.
Compact Sequence Notation
The notation
\[ (a_n)_{n=1}^{\infty} \]means that the sequence has terms
\[ a_1,\ a_2,\ a_3,\ldots \]and continues indefinitely.
Similarly,
\[ (a_n)_{n=0}^{\infty} \]means
\[ a_0,\ a_1,\ a_2,\ldots \]This notation records both the sequence name and its indexing convention.
Finite Sequences
A finite sequence contains a specified finite number of terms.
For example,
\[ 3,\ 6,\ 9,\ 12,\ 15 \]contains five terms.
We could write
\[ (a_n)_{n=1}^{5} \]where
\[ a_n=3n. \]The final term is
\[ a_5=15. \]Infinite Sequences
An infinite sequence has no final index.
For example,
\[ 1,\ \frac12,\ \frac14,\ \frac18,\ldots \]continues indefinitely.
If it begins at \(n=1\), one possible formula is
\[ a_n=\frac{1}{2^{n-1}}. \]The notation
\[ (a_n)_{n=1}^{\infty} \]indicates that there are infinitely many indexed terms.
Finite Does Not Mean Bounded
The words finite and infinite describe the number of terms, not their numerical size.
The finite sequence
\[ 1,\ 10,\ 100,\ 1000,\ 1000000 \]has only five terms even though its final value is large.
Meanwhile,
\[ 1,\ \frac12,\ \frac13,\ \frac14,\ldots \]has infinitely many terms even though all of them remain between \(0\) and \(1\).
Finding a Particular Term from a Formula
Suppose
\[ a_n=n^2+2. \]To find \(a_5\), substitute
\[ n=5. \]Then
\[ a_5=5^2+2 \] \[ =25+2 \] \[ =\boxed{27}. \]Evaluating a sequence formula works much like evaluating any other function.
Example: Several Terms from One Rule
Let
\[ a_n=2n^2-1 \]for \(n\geq1\).
The first term is
\[ a_1=2(1)^2-1=1. \]The second term is
\[ a_2=2(2)^2-1=7. \]The third term is
\[ a_3=2(3)^2-1=17. \]The fourth term is
\[ a_4=2(4)^2-1=31. \]Therefore, the sequence begins
\[ 1,\ 7,\ 17,\ 31,\ldots \]Indices Can Appear Inside More Complicated Expressions
Once \(a_n\) represents the term at index \(n\), nearby terms can be written relative to \(n\).
The notation
\[ a_{n+1} \]means the term one index after \(a_n\).
Similarly,
\[ a_{n-1} \]means the term one index before \(a_n\), whenever that index belongs to the sequence.
These shifted indices will become central when we study recursive definitions.
Example: Evaluating a Shifted Index
Suppose
\[ a_n=3n+2. \]Then
\[ a_{n+1} \]is found by replacing every occurrence of \(n\) in the rule with \(n+1\):
\[ a_{n+1}=3(n+1)+2. \]Simplify:
\[ a_{n+1}=3n+5. \]Similarly,
\[ a_{n-1}=3(n-1)+2 \] \[ =3n-1. \]Be Careful with \(a_{n+1}\)
The notation
\[ a_{n+1} \]means the term whose index is \(n+1\).
It does not generally mean
\[ a_n+1. \]For example, if
\[ a_n=n^2, \]then
\[ a_{n+1}=(n+1)^2 \] \[ =n^2+2n+1, \]while
\[ a_n+1=n^2+1. \]These are different expressions.
Index and Value Are Different Ideas
One of the most important distinctions in sequence work is the difference between a term's index and its value.
Suppose
\[ a_n=5n. \]Then
\[ a_4=20. \]Here,
- the index is \(4\);
- the term value is \(20\).
The index tells us where the term occurs. The formula tells us the value associated with that index.
A Sequence Table
Sequence information can be organized in a table.
For
\[ a_n=2n+1, \]we obtain:
| Index \(n\) | Term \(a_n\) |
|---|---|
| \(1\) | \(3\) |
| \(2\) | \(5\) |
| \(3\) | \(7\) |
| \(4\) | \(9\) |
| \(5\) | \(11\) |
The left column contains the inputs. The right column contains the corresponding sequence values.
Sequence Terms as Ordered Pairs
The same sequence can be represented by ordered pairs
\[ (n,a_n). \]For
\[ a_n=2n+1, \]the first few pairs are
\[ (1,3),\quad (2,5),\quad (3,7),\quad (4,9),\quad (5,11). \]This representation prepares us to graph a sequence in the coordinate plane.
Graphing a Sequence
A numerical sequence can be graphed by plotting points
\[ (n,a_n). \]For example, the sequence
\[ a_n=2n+1 \]produces the points
\[ (1,3),\ (2,5),\ (3,7),\ (4,9),\ldots \]Because the allowed indices are discrete, these points are normally plotted separately rather than joined into a continuous curve.
The gaps between the plotted points are meaningful: there may be no sequence term associated with the intermediate horizontal coordinates.
A Formula May Describe More Than the Sequence
Consider the formula
\[ y=2x+1. \]As an ordinary real-valued function, this formula may describe every real input \(x\).
But the sequence
\[ a_n=2n+1,\qquad n=1,2,3,\ldots \]uses only the specified integer indices.
The algebraic rule may be the same, but the domains are different.
A formula and its domain must therefore be considered together.
Position Language and Index Language
When indexing begins at \(1\), ordinary position language and index language match naturally:
\[ \text{first term}=a_1, \] \[ \text{second term}=a_2, \] \[ \text{third term}=a_3. \]But when indexing begins at \(0\), the first listed term is
\[ a_0. \]This is why precise mathematical writing often refers directly to the term at index \(n\) rather than relying only on words such as first or second.
Indexing from \(0\): A Common Source of Confusion
Suppose
\[ (a_n)_{n=0}^{\infty} \]begins
\[ 5,\ 8,\ 11,\ 14,\ldots \]Then
\[ a_0=5, \] \[ a_1=8, \] \[ a_2=11, \]and
\[ a_3=14. \]The term \(8\) is the second displayed term, but its index is \(1\).
Terms Need Not Be Positive
Sequence terms can be negative, zero, positive, or mixtures of these.
For example,
\[ -3,\ -1,\ 1,\ 3,\ 5,\ldots \]is a sequence.
If indexing begins at \(1\), one possible rule is
\[ a_n=2n-5. \]Then
\[ a_1=-3,\qquad a_2=-1,\qquad a_3=1. \]Terms Need Not Be Integers
The indices of a standard sequence are often integers, but the term values can be many kinds of mathematical objects.
For example,
\[ 1,\ \frac12,\ \frac13,\ \frac14,\ldots \]contains rational numbers.
The sequence
\[ \sqrt{1},\ \sqrt{2},\ \sqrt{3},\ \sqrt{4},\ldots \]contains both rational and irrational values.
The indexing structure and the nature of the term values are separate questions.
Sequences Can Contain Other Mathematical Objects
Although introductory sequences often contain numbers, the underlying idea is broader.
A sequence may contain:
- numbers;
- points;
- vectors;
- matrices;
- functions;
- sets;
- symbols;
- other mathematical objects.
What makes the collection a sequence is that its members are indexed in an ordered way.
Example: A Sequence of Points
Suppose
\[ P_n=(n,n^2). \]Then
\[ P_1=(1,1), \] \[ P_2=(2,4), \]and
\[ P_3=(3,9). \]The terms of this sequence are ordered pairs rather than single numbers.
The notation and indexing principles remain the same.
Changing the Index Can Change the Formula
The same displayed values can be indexed in different ways.
Consider
\[ 5,\ 8,\ 11,\ 14,\ldots \]If the first term is \(a_1\), then one formula is
\[ a_n=3n+2. \]But if the first term is \(b_0\), then one formula is
\[ b_n=3n+5. \]Both formulas generate the same displayed values in the same order, but they attach different indices to them.
Index Shifts
Changing where a sequence begins is often called an index shift.
For example,
\[ a_n=3n+2,\qquad n\geq1 \]produces
\[ 5,\ 8,\ 11,\ 14,\ldots \]while
\[ b_n=3n+5,\qquad n\geq0 \]produces the same values.
Index shifts appear throughout algebra, combinatorics, calculus, computer science, and recurrence relations.
Identifying Terms from a Listed Sequence
Suppose
\[ a_1=7,\quad a_2=12,\quad a_3=17,\quad a_4=22,\quad a_5=27. \]Then
\[ a_3=17 \]and
\[ a_5=27. \]If asked for the index at which the value \(22\) occurs, the answer is
\[ n=4. \]Reading sequence notation requires moving comfortably in both directions: from index to value and from value to index.
A Value May Occur at More Than One Index
Suppose
\[ 2,\ 5,\ 2,\ 8,\ 2,\ldots \]is indexed beginning at \(1\).
Then
\[ a_1=2,\qquad a_3=2,\qquad a_5=2. \]Therefore, asking
“At what index does the value \(2\) occur?”
may have more than one answer.
Sequences do not require different indices to have different values.
Constant Sequences
A sequence may assign the same value to every index.
For example,
\[ 6,\ 6,\ 6,\ 6,\ldots \]can be defined by
\[ a_n=6. \]Thus,
\[ a_1=a_2=a_3=\cdots=6. \]A changing index does not require a changing term value.
Alternating Sequences
Terms may also alternate between values.
Consider
\[ -1,\ 1,\ -1,\ 1,\ -1,\ldots \]beginning at \(n=1\).
One formula is
\[ a_n=(-1)^n. \]Indeed,
\[ a_1=-1,\qquad a_2=1,\qquad a_3=-1,\qquad a_4=1. \]Exponents involving the index are frequently used to encode alternating behavior.
The Index Can Appear in Many Kinds of Formulas
Sequence formulas can involve familiar algebraic operations.
Examples include
\[ a_n=4n-1, \] \[ b_n=n^2+3n, \] \[ c_n=2^n, \] \[ d_n=\frac{1}{n}, \]and
\[ e_n=(-1)^n n. \]Each rule assigns a term value to an allowed index.
Later lessons will classify important families of sequences generated by different kinds of rules.
Not Every Sequence Is Given by a Simple Formula
Some sequences have compact formulas for \(a_n\), but others are more naturally described by another method.
A sequence may be specified by:
- listing its terms;
- giving a formula for the \(n\)th term;
- describing how each term is generated from earlier terms;
- providing an algorithm;
- giving a table or other structured rule.
The next lesson will distinguish explicit definitions from recursive definitions in detail.
Index Restrictions Matter
Suppose
\[ a_n=\frac{1}{n-3}. \]The expression is undefined at
\[ n=3. \]Therefore, if the intended indexing set includes \(3\), the sequence rule requires additional clarification or restriction.
As with ordinary functions, a valid sequence definition must assign an appropriate value at every allowed index.
Example: A Restricted Sequence
We could define
\[ a_n=\frac{1}{n-3} \]for
\[ n=4,5,6,\ldots \]Then
\[ a_4=1, \] \[ a_5=\frac12, \] \[ a_6=\frac13. \]Thus the sequence begins
\[ 1,\ \frac12,\ \frac13,\ldots \]The indexing restriction is part of the definition.
Reading Sequence Notation Carefully
Suppose
\[ a_n=n^2-1,\qquad n\geq0. \]From this single statement we can identify three pieces of information:
- the sequence is named \(a\);
- the value at index \(n\) is \(n^2-1\);
- the allowed indices begin at \(0\).
Therefore,
\[ a_0=-1,\qquad a_1=0,\qquad a_2=3,\qquad a_3=8. \]The sequence begins
\[ -1,\ 0,\ 3,\ 8,\ldots \]Common Error: Treating the Index as the Term
If
\[ a_n=4n+1, \]then \(a_3\) does not equal \(3\).
The number \(3\) is the index.
The term value is
\[ a_3=4(3)+1 \] \[ =\boxed{13}. \]Always distinguish the input label from the output value.
Common Error: Assuming Every Sequence Begins at \(1\)
Suppose
\[ a_n=n^2,\qquad n\geq0. \]The first indexed term is
\[ a_0=0, \]not
\[ a_1=1. \]Before evaluating or listing terms, always check the stated starting index.
Common Error: Confusing \(a_{n+1}\) with \(a_n+1\)
If
\[ a_n=2n, \]then
\[ a_{n+1}=2(n+1)=2n+2. \]But
\[ a_n+1=2n+1. \]Therefore,
\[ a_{n+1}\neq a_n+1 \]in this example.
The subscript belongs to the index of the sequence term.
Common Error: Joining Discrete Sequence Points Automatically
If a sequence is defined only for
\[ n=1,2,3,\ldots, \]then its graph consists of points at those allowed indices.
Drawing a continuous line between them can falsely suggest that values such as
\[ a_{1.5} \]are part of the sequence.
Unless a continuous extension has been separately defined, graph the sequence discretely.
Common Error: Assuming an Ellipsis Defines a Unique Rule
A list such as
\[ 1,\ 4,\ 9,\ 16,\ldots \]strongly suggests the square numbers.
A natural formula is
\[ a_n=n^2. \]But the displayed terms alone do not logically force every later term.
In formal work, the defining rule or pattern should be stated whenever precision matters.
A Reliable Procedure for Reading a Sequence
- Identify the name of the sequence.
- Identify the allowed indices.
- Determine the starting index.
- Identify how the term value is specified.
- Keep the index separate from the term value.
- Substitute the requested index when a formula is given.
- Check whether the sequence is finite or infinite.
- Preserve the order of the terms.
A Reliable Procedure for Generating Terms
If an explicit formula \(a_n\) is given:
- determine the first allowed value of \(n\);
- substitute that index into the formula;
- simplify to obtain the term value;
- move to the next allowed index;
- repeat until the required terms have been generated.
For
\[ a_n=3n-2,\qquad n\geq1, \]this produces
\[ 1,\ 4,\ 7,\ 10,\ldots \]Example: Full Notation Workflow
Consider
\[ b_n=n^2+2n,\qquad n=0,1,2,3,\ldots \]First evaluate \(b_0\):
\[ b_0=(0)^2+2(0)=0. \]Then
\[ b_1=(1)^2+2(1)=3, \] \[ b_2=(2)^2+2(2)=8, \]and
\[ b_3=(3)^2+2(3)=15. \]Therefore, the sequence begins
\[ \boxed{0,\ 3,\ 8,\ 15,\ldots}. \]The term at index \(3\) is
\[ \boxed{b_3=15}. \]Example: Working Backward from a Term Value
Suppose
\[ a_n=4n+3,\qquad n\geq1. \]At what index does the value \(35\) occur?
Set the general term equal to \(35\):
\[ 4n+3=35. \]Subtract \(3\):
\[ 4n=32. \]Divide by \(4\):
\[ n=8. \]Therefore,
\[ \boxed{a_8=35}. \]Example: Does a Value Occur in the Sequence?
Let
\[ a_n=3n+1,\qquad n\geq1. \]Does the value \(20\) occur?
Solve
\[ 3n+1=20. \]Then
\[ 3n=19 \]and
\[ n=\frac{19}{3}. \]But the allowed indices are positive integers.
Therefore, \(20\) is not a term of this sequence.
Why Sequence Notation Matters
Sequence notation gives us a precise language for describing quantities that evolve step by step.
It allows us to ask questions such as:
- What is the term at a particular position?
- How does one term compare with the next?
- Can the \(n\)th term be calculated directly?
- Can later terms be generated from earlier ones?
- What pattern appears in the differences between terms?
- What happens to the terms as the index becomes large?
- What happens when many terms are added together?
These questions lead directly to recurrence relations, arithmetic and geometric sequences, finite differences, series, and convergence.
Practice 1: Identify the Terms
Consider the sequence
\[ 4,\ 9,\ 14,\ 19,\ 24. \]If indexing begins at \(1\), find
\[ a_1,\qquad a_3,\qquad a_5. \]Practice 2: Evaluate a Term
Let
\[ a_n=5n-2,\qquad n\geq1. \]Find
\[ a_7. \]Practice 3: Generate Terms
Let
\[ b_n=n^2+1,\qquad n\geq1. \]Write the first five terms.
Practice 4: Indexing from Zero
Let
\[ c_n=2n+5,\qquad n\geq0. \]Find
\[ c_0,\qquad c_1,\qquad c_4. \]Practice 5: Shifted Index
If
\[ a_n=4n-3, \]find a simplified expression for
\[ a_{n+1}. \]Practice 6: Previous Index
If
\[ a_n=n^2+2n, \]simplify
\[ a_{n-1}. \]Practice 7: Find the Index
Suppose
\[ a_n=6n+1,\qquad n\geq1. \]At what index does
\[ a_n=49 \]occur?
Practice 8: Is It a Term?
Let
\[ a_n=4n+2,\qquad n\geq1. \]Determine whether \(31\) is a term of the sequence.
Practice 9: Finite or Infinite?
Classify each sequence as finite or infinite.
A.
\[ (a_n)_{n=1}^{20} \]B.
\[ (b_n)_{n=0}^{\infty} \]Practice 10: Identify the Error
A student is given
\[ a_n=n^2,\qquad n\geq0 \]and writes
\[ 1,\ 4,\ 9,\ 16,\ldots \]Explain the error.
Practice Answers
Practice 1
The first, third, and fifth terms are
\[ \boxed{ a_1=4,\qquad a_3=14,\qquad a_5=24 }. \]Practice 2
\[ a_7=5(7)-2 \] \[ =35-2 \] \[ =\boxed{33}. \]Practice 3
\[ b_1=1^2+1=2, \] \[ b_2=2^2+1=5, \] \[ b_3=3^2+1=10, \] \[ b_4=4^2+1=17, \]and
\[ b_5=5^2+1=26. \]Therefore,
\[ \boxed{2,\ 5,\ 10,\ 17,\ 26}. \]Practice 4
\[ c_0=2(0)+5=5, \] \[ c_1=2(1)+5=7, \]and
\[ c_4=2(4)+5=13. \]Thus,
\[ \boxed{ c_0=5,\qquad c_1=7,\qquad c_4=13 }. \]Practice 5
Replace \(n\) with \(n+1\):
\[ a_{n+1}=4(n+1)-3 \] \[ =4n+4-3 \] \[ =\boxed{4n+1}. \]Practice 6
Replace \(n\) with \(n-1\):
\[ a_{n-1}=(n-1)^2+2(n-1). \]Expand:
\[ a_{n-1}=n^2-2n+1+2n-2. \]Therefore,
\[ \boxed{a_{n-1}=n^2-1}. \]Practice 7
Solve
\[ 6n+1=49. \]Then
\[ 6n=48 \]and
\[ \boxed{n=8}. \]Therefore,
\[ a_8=49. \]Practice 8
Set
\[ 4n+2=31. \]Then
\[ 4n=29 \]and
\[ n=\frac{29}{4}. \]This is not an allowed positive-integer index.
Therefore,
\[ \boxed{31\text{ is not a term of the sequence}.} \]Practice 9
Sequence A ends at index \(20\), so it is
\[ \boxed{\text{finite}}. \]Sequence B continues through arbitrarily large indices, so it is
\[ \boxed{\text{infinite}}. \]Practice 10
The indexing begins at \(n=0\), so the first term must be
\[ a_0=0^2=0. \]The sequence should begin
\[ \boxed{0,\ 1,\ 4,\ 9,\ 16,\ldots}. \]The student incorrectly began evaluating at \(n=1\).
What to Remember
A sequence is an ordered collection of terms.
- Each term has an index.
- \(a_n\) means the term of sequence \(a\) at index \(n\).
- The index and the term value are different quantities.
- A sequence may begin at \(0\), \(1\), or another specified index.
- The indexing set is part of the sequence definition.
- A sequence may be finite or infinite.
- Repeated values are allowed.
- Order matters.
- A sequence can be viewed as a function defined on a discrete index set.
- \(a_{n+1}\) denotes a shifted index, not \(a_n+1\).
The fundamental notation is
\[ \boxed{n\longmapsto a_n}. \]The index identifies a position, and \(a_n\) gives the object assigned to that position.
Where We Go Next
We now have the language needed to identify sequence terms and their positions. The next question is how an entire sequence can be defined efficiently.
The next lesson, Explicit and Recursive Definitions, compares two major methods for specifying sequences: calculating a term directly from its index and generating a term from one or more earlier terms.