The Central Idea
A sequence is an ordered collection of terms:
\[ a_1,\ a_2,\ a_3,\ a_4,\ldots \]The subscript records position. Thus \(a_1\) is the first term, \(a_2\) is the second term, and \(a_n\) represents the term in position \(n\).
There are two especially important ways to specify a sequence.
- An explicit rule gives \(a_n\) directly from the index \(n\).
- A recursive rule gives one or more starting terms and a rule for obtaining later terms from earlier ones.
The distinction is about how information is organized, not about whether the sequence itself is different.
Sequences Are Ordered
Order matters in a sequence.
For example,
\[ 2,\ 4,\ 6,\ 8,\ldots \]and
\[ 8,\ 6,\ 4,\ 2,\ldots \]contain some of the same values, but they are not the same sequence because the values occur in different positions.
A sequence therefore connects two pieces of information:
\[ \boxed{\text{position} \longrightarrow \text{term value}}. \]This is why sequence notation resembles function notation.
Sequence Notation
A common notation for a sequence is
\[ \{a_n\}_{n=1}^{\infty}. \]This means that the terms are indexed by
\[ n=1,2,3,\ldots \]and that the value in position \(n\) is called \(a_n\).
We may also write the sequence simply as
\[ a_1,a_2,a_3,\ldots \]or list several terms before an ellipsis.
The Index
The number \(n\) used to identify a term is called an index.
In
\[ a_n=3n+1, \]\(n\) is the index.
If the sequence begins at \(n=1\), then
\[ a_1=3(1)+1=4, \] \[ a_2=3(2)+1=7, \] \[ a_3=3(3)+1=10. \]Therefore, the sequence begins
\[ \boxed{4,\ 7,\ 10,\ldots}. \]Indexing Does Not Always Begin at \(1\)
Many elementary sequences begin with \(a_1\), but indexing can begin at another integer.
Suppose
\[ b_n=2^n,\qquad n\ge 0. \]Then the first listed term is
\[ b_0=2^0=1. \]The next terms are
\[ b_1=2,\qquad b_2=4,\qquad b_3=8. \]Thus,
\[ \boxed{1,\ 2,\ 4,\ 8,\ldots}. \]Always inspect the stated starting index before evaluating a sequence.
What Is an Explicit Sequence Rule?
An explicit rule gives the value of a term directly from its index.
For example,
\[ \boxed{a_n=5n-2}. \]To find \(a_{100}\), we substitute \(n=100\):
\[ a_{100}=5(100)-2=498. \]We do not need to calculate the first \(99\) terms.
Explicit Rules Behave Like Formulas
An explicit rule usually has the form
\[ \boxed{a_n=f(n)}. \]The index \(n\) is the input, and the term value \(a_n\) is the output.
For
\[ a_n=n^2+2, \]we have
\[ a_1=3,\qquad a_2=6,\qquad a_3=11,\qquad a_4=18. \]So the sequence begins
\[ \boxed{3,\ 6,\ 11,\ 18,\ldots}. \]Evaluating an Explicit Rule
Suppose
\[ a_n=2n^2-3n+4. \]To find \(a_5\), substitute \(n=5\):
\[ a_5=2(5)^2-3(5)+4. \]Simplify:
\[ a_5=50-15+4=39. \]Therefore,
\[ \boxed{a_5=39}. \]Explicit Rules Can Produce Alternating Behavior
Explicit rules are not limited to steadily increasing or decreasing sequences.
Consider
\[ a_n=(-1)^n. \]Then
\[ a_1=-1,\qquad a_2=1,\qquad a_3=-1,\qquad a_4=1. \]Thus,
\[ \boxed{-1,\ 1,\ -1,\ 1,\ldots}. \]The factor \((-1)^n\) is a common way to encode alternating signs.
Explicit Rules Can Use Fractions
Consider
\[ a_n=\frac{1}{n}. \]The first several terms are
\[ 1,\ \frac12,\ \frac13,\ \frac14,\ldots \]and the \(100\)th term is immediately available:
\[ \boxed{a_{100}=\frac{1}{100}}. \]This direct access to distant terms is one of the main advantages of an explicit representation.
Explicit Rules Can Be Piecewise
A sequence may follow different formulas for different indices.
For example,
\[ a_n= \begin{cases} n, & n \text{ odd},\\ 2n, & n \text{ even}. \end{cases} \]Then
\[ a_1=1,\qquad a_2=4,\qquad a_3=3,\qquad a_4=8. \]Therefore, the sequence begins
\[ \boxed{1,\ 4,\ 3,\ 8,\ldots}. \]What Is a Recursive Sequence Rule?
A recursive rule defines later terms using one or more earlier terms.
For example,
\[ a_1=4, \] \[ \boxed{a_n=a_{n-1}+3\qquad (n\ge 2)}. \]The rule says: start at \(4\), then add \(3\) to the preceding term.
Therefore,
\[ a_2=4+3=7, \] \[ a_3=7+3=10, \] \[ a_4=10+3=13. \]The sequence begins
\[ \boxed{4,\ 7,\ 10,\ 13,\ldots}. \]A Recursive Rule Needs a Starting Point
The relation
\[ a_n=a_{n-1}+5 \]does not identify a unique sequence by itself.
If \(a_1=1\), then the sequence begins
\[ 1,\ 6,\ 11,\ 16,\ldots \]But if \(a_1=10\), then it begins
\[ 10,\ 15,\ 20,\ 25,\ldots \]The recursive relationship describes the step-to-step change, while the starting value determines where the sequence begins.
Initial Values and Recursive Rules Work Together
A recursive sequence usually contains two ingredients:
- one or more initial values;
- a recurrence telling how to obtain later terms.
For example,
\[ a_1=2, \] \[ a_n=2a_{n-1}\qquad(n\ge 2) \]gives
\[ a_2=2(2)=4, \] \[ a_3=2(4)=8, \] \[ a_4=2(8)=16. \]Thus,
\[ \boxed{2,\ 4,\ 8,\ 16,\ldots}. \]Reading the Previous-Term Notation
In a recursive rule, the expression
\[ a_{n-1} \]means the term immediately before \(a_n\).
If \(n=6\), then
\[ a_{n-1}=a_5. \]Therefore, a rule such as
\[ a_n=a_{n-1}+4 \]tells us that
\[ a_6=a_5+4. \]The notation encodes a dependency between neighboring positions.
Recursion Can Depend on More Than One Earlier Term
A recursive rule may use several earlier terms.
For example,
\[ a_1=1,\qquad a_2=1, \] \[ \boxed{a_n=a_{n-1}+a_{n-2}\qquad(n\ge 3)}. \]Then
\[ a_3=1+1=2, \] \[ a_4=2+1=3, \] \[ a_5=3+2=5. \]The sequence begins
\[ \boxed{1,\ 1,\ 2,\ 3,\ 5,\ldots}. \]Later lessons will study this type of multi-term recurrence in greater depth.
Why More Initial Values May Be Needed
If a recurrence uses two previous terms, then two starting values are usually needed.
In
\[ a_n=a_{n-1}+a_{n-2}, \]calculating \(a_3\) requires both
\[ a_2 \]and
\[ a_1. \]One starting value is not enough to determine \(a_3\).
More generally, a recursive rule must be supplied with enough initial information to begin the process unambiguously.
Explicit and Recursive Descriptions of the Same Sequence
The sequence
\[ 5,\ 8,\ 11,\ 14,\ldots \]can be described recursively by
\[ a_1=5, \] \[ a_n=a_{n-1}+3\qquad(n\ge 2). \]It can also be described explicitly by
\[ \boxed{a_n=5+3(n-1)}. \]Simplifying,
\[ \boxed{a_n=3n+2}. \]These rules describe the same sequence from different viewpoints.
Position View versus Process View
An explicit rule emphasizes position.
\[ \boxed{n \longrightarrow a_n} \]A recursive rule emphasizes process.
\[ \boxed{\text{earlier term or terms} \longrightarrow \text{next term}} \]Neither viewpoint is automatically better. The useful representation depends on the question being asked.
Finding a Distant Term: Explicit Form
Suppose
\[ a_n=7+4(n-1). \]To find the \(50\)th term,
\[ a_{50}=7+4(49). \]Therefore,
\[ a_{50}=7+196=203. \]So
\[ \boxed{a_{50}=203}. \]The explicit form gives direct access to the requested position.
Finding a Distant Term: Recursive Form
Suppose the same sequence is given by
\[ a_1=7, \] \[ a_n=a_{n-1}+4. \]We can generate
\[ 7,\ 11,\ 15,\ 19,\ldots \]one step at a time.
This is natural when we want to model a repeated process, but finding \(a_{50}\) directly is less convenient unless we first obtain an explicit formula or use an efficient computational procedure.
Recursive Rules Can Model Repeated Change Naturally
Suppose a quantity begins at \(100\) and increases by \(8\) after each stage.
If \(a_n\) records the quantity after stage \(n\), a recursive description might be
\[ a_0=100, \] \[ \boxed{a_n=a_{n-1}+8\qquad(n\ge 1)}. \]The recurrence mirrors the repeated instruction: take the previous amount and add \(8\).
This process-oriented description is often the most natural one when the system itself evolves step by step.
Explicit Rules Can Make Global Patterns Easier to See
The recursive sequence
\[ a_1=2,\qquad a_n=a_{n-1}+5 \]begins
\[ 2,\ 7,\ 12,\ 17,\ldots \]Its explicit rule is
\[ a_n=2+5(n-1). \]Simplifying,
\[ \boxed{a_n=5n-3}. \]The explicit form reveals immediately that term values depend linearly on the index \(n\).
From a Simple Recursive Rule to an Explicit Rule
Consider
\[ a_1=6, \] \[ a_n=a_{n-1}+4. \]Each new term adds \(4\).
To move from the first term to the \(n\)th term requires
\[ n-1 \]additions of \(4\).
Therefore,
\[ a_n=6+4(n-1). \]Hence,
\[ \boxed{a_n=4n+2}. \]Why the Factor \(n-1\) Appears
If a sequence begins at \(a_1\), then no repeated step has occurred at the first term.
One step is needed to reach \(a_2\), two steps to reach \(a_3\), and so on.
Therefore, the number of transitions from \(a_1\) to \(a_n\) is
\[ \boxed{n-1}. \]This is why a constant-step recursive sequence
\[ a_n=a_{n-1}+d \]corresponds to the explicit structure
\[ a_n=a_1+(n-1)d. \]A later lesson develops arithmetic recurrences systematically.
From an Explicit Rule to a Recursive Rule
Suppose
\[ a_n=4n+1. \]First find the initial term:
\[ a_1=4(1)+1=5. \]Now compare consecutive terms.
\[ a_{n-1}=4(n-1)+1=4n-3. \]Therefore,
\[ a_n-a_{n-1} =(4n+1)-(4n-3) =4. \]So an equivalent recursive description is
\[ \boxed{a_1=5,\qquad a_n=a_{n-1}+4}. \]Recursive Rules Need Their Valid Index Range
Consider
\[ a_1=3,\qquad a_n=a_{n-1}+2. \]The recurrence cannot be applied at \(n=1\), because that would require \(a_0\), which has not been defined.
A complete statement is therefore
\[ \boxed{a_1=3,\qquad a_n=a_{n-1}+2\quad(n\ge 2)}. \]The index condition is part of the definition.
A Sequence Can Be Finite
Not every sequence continues forever.
The finite sequence
\[ 2,\ 5,\ 8,\ 11,\ 14 \]can be defined explicitly by
\[ a_n=3n-1,\qquad 1\le n\le 5. \]The index restriction tells us that the sequence stops after \(a_5\).
A formula alone does not always tell us whether the intended sequence is finite or infinite; the domain of allowable indices matters.
A Sequence Can Be Infinite
An infinite sequence continues without a final term.
For example,
\[ a_n=\frac{1}{n},\qquad n\ge 1 \]defines
\[ 1,\ \frac12,\ \frac13,\ \frac14,\ldots \]The ellipsis indicates continuation according to the stated rule.
Tables Help Connect Indices and Terms
For
\[ a_n=n^2+1, \]a short table is
| \(n\) | \(a_n\) |
|---|---|
| \(1\) | \(2\) |
| \(2\) | \(5\) |
| \(3\) | \(10\) |
| \(4\) | \(17\) |
A table makes the input-output structure visible without changing the underlying definition.
Sequences as Discrete Functions
A sequence can be viewed as a function whose inputs are discrete index values such as
\[ 1,2,3,4,\ldots \]rather than every real number in an interval.
Under this viewpoint,
\[ a_n \]plays a role similar to
\[ f(n). \]The sequence
\[ a_n=2n+3 \]therefore associates each allowed integer index with a term value.
Graphs of Sequences Are Discrete
If we graph a sequence, we plot points
\[ (n,a_n). \]For
\[ a_n=2n+1,\qquad n\ge 1, \]the first points are
\[ (1,3),\ (2,5),\ (3,7),\ (4,9). \]These points lie on the line \(y=2x+1\), but the sequence consists only of the allowed indexed points—not every point on the continuous line.
A Recursive Rule May Be Easier to Discover
Suppose we observe
\[ 3,\ 6,\ 12,\ 24,\ldots \]The repeated step is immediately visible:
\[ \text{multiply the previous term by }2. \]Therefore, a natural recursive rule is
\[ \boxed{a_1=3,\qquad a_n=2a_{n-1}}. \]An explicit formula also exists,
\[ a_n=3\cdot2^{n-1}, \]but the recursive pattern may be easier to notice first.
An Explicit Rule May Be Easier to Use
For the same sequence,
\[ a_n=3\cdot2^{n-1}, \]the \(20\)th term can be obtained directly:
\[ a_{20}=3\cdot2^{19}. \]The explicit rule avoids generating all preceding terms.
This illustrates an important practical distinction: recursive rules often describe how a sequence grows, while explicit rules often make distant-term evaluation easier.
Not Every Recursive Rule Has a Simple Explicit Form
Some recurrences can be converted easily to familiar formulas.
Others require more advanced methods, and some do not produce a simple elementary closed form.
For example, a recurrence involving several earlier terms can be easy to compute recursively even when its direct formula is much less obvious.
Later lessons introduce closed forms, characteristic methods, generating functions, and other tools for studying these structures.
Not Every Listed Pattern Determines a Unique Rule
Suppose we are shown only
\[ 1,\ 2,\ 4,\ 8. \]A natural guess is that each term doubles, giving
\[ 1,\ 2,\ 4,\ 8,\ 16,\ldots \]But finitely many listed terms do not logically force a unique continuation. Many different rules can agree on the first four terms and diverge later.
When identifying a sequence from data, the intended pattern must be stated or justified rather than treated as mathematically inevitable.
Example: Evaluate Terms from an Explicit Rule
Let
\[ a_n=3n^2-2. \]Find \(a_2\) and \(a_6\).
For \(n=2\),
\[ a_2=3(2)^2-2=12-2=10. \]For \(n=6\),
\[ a_6=3(6)^2-2=108-2=106. \]Therefore,
\[ \boxed{a_2=10,\qquad a_6=106}. \]Example: Generate Terms from a Recursive Rule
Let
\[ a_1=10, \] \[ a_n=a_{n-1}-3\qquad(n\ge 2). \]Then
\[ a_2=10-3=7, \] \[ a_3=7-3=4, \] \[ a_4=4-3=1, \] \[ a_5=1-3=-2. \]Thus, the first five terms are
\[ \boxed{10,\ 7,\ 4,\ 1,\ -2}. \]Example: Translate a Verbal Process into Recursion
A quantity starts at \(50\). At each stage, multiply the previous value by \(1.2\).
Let \(a_n\) represent the value after \(n\) stages, with \(a_0=50\).
The recursive description is
\[ \boxed{a_0=50,\qquad a_n=1.2a_{n-1}\quad(n\ge 1)}. \]The first few values are
\[ 50,\ 60,\ 72,\ 86.4,\ldots \]The recurrence directly mirrors the repeated operation.
Example: Translate a Verbal Position Rule into Explicit Form
Suppose the \(n\)th term is three more than twice its position.
"Twice its position" is
\[ 2n, \]so three more gives
\[ \boxed{a_n=2n+3}. \]The first terms are
\[ 5,\ 7,\ 9,\ 11,\ldots \]Example: Compare Two Representations
Consider the explicit rule
\[ a_n=2^{n-1}. \]Its first terms are
\[ 1,\ 2,\ 4,\ 8,\ 16,\ldots \]Each term after the first is twice the preceding term.
Therefore, an equivalent recursive rule is
\[ \boxed{a_1=1,\qquad a_n=2a_{n-1}\quad(n\ge 2)}. \]Explicit Form Is Not the Same as Listing Terms
Writing
\[ 2,\ 5,\ 8,\ 11,\ldots \]displays terms, but it does not by itself state an explicit formula.
An explicit formula for the intended constant-step pattern is
\[ \boxed{a_n=3n-1}. \]The formula explains how to compute a term from its index.
Recursive Form Is Not Just a Pattern Description
Saying
"add \(4\) each time"
describes a repeated action but is incomplete as a formal recursive definition unless the starting value is also known.
A complete recursive definition might be
\[ \boxed{a_1=7,\qquad a_n=a_{n-1}+4\quad(n\ge 2)}. \]Common Error: Ignoring the Starting Index
Suppose
\[ a_n=n^2,\qquad n\ge 0. \]A student lists
\[ 1,\ 4,\ 9,\ldots \]as the beginning of the sequence.
This skips the term at \(n=0\):
\[ a_0=0^2=0. \]The correct beginning is
\[ \boxed{0,\ 1,\ 4,\ 9,\ldots}. \]Common Error: Treating \(a_n\) as Multiplication
The notation
\[ a_n \]means "the term indexed by \(n\)." It does not normally mean
\[ a\cdot n. \]The subscript is part of the term's label.
Common Error: Substituting into the Subscript Incorrectly
Suppose
\[ a_n=2n+5. \]To find \(a_7\), replace \(n\) by \(7\):
\[ a_7=2(7)+5=19. \]The subscript tells which index value to use in the formula.
Common Error: Forgetting an Initial Value
The rule
\[ a_n=2a_{n-1} \]does not specify a unique sequence without a starting term.
Both
\[ 1,\ 2,\ 4,\ 8,\ldots \]and
\[ 3,\ 6,\ 12,\ 24,\ldots \]satisfy the same recurrence.
The initial value distinguishes them.
Common Error: Applying a Recurrence Too Early
Suppose
\[ a_1=4,\qquad a_n=a_{n-1}+2\quad(n\ge 2). \]The recursive formula begins at \(n=2\), not at \(n=1\).
At \(n=2\),
\[ a_2=a_1+2=6. \]The initial term \(a_1\) is supplied directly rather than generated by the recurrence.
Common Error: Using the Wrong Previous Term
In
\[ a_n=a_{n-1}+5, \]finding \(a_6\) requires \(a_5\), not \(a_4\).
The subscript
\[ n-1 \]always means one index before \(n\).
Common Error: Assuming the Next Term Is Certain from a Short List
A short list such as
\[ 2,\ 4,\ 6,\ 8 \]strongly suggests a constant increase of \(2\), but those four terms alone do not prove that the next term must be \(10\).
The intended rule should be given, inferred from context, or stated as an assumption.
Common Error: Confusing Term Number with Term Value
In the sequence
\[ 10,\ 20,\ 30,\ 40,\ldots \]the third term is
\[ a_3=30. \]The number \(3\) is the position; the number \(30\) is the value in that position.
A Reliable Procedure for Explicit Sequences
- Identify the explicit rule for \(a_n\).
- Check the allowed index values.
- Substitute the requested index for \(n\).
- Evaluate using the usual order of operations.
- Interpret the result as the term value at that position.
- When listing terms, repeat the process for each requested index.
A Reliable Procedure for Recursive Sequences
- Identify all initial values.
- Identify the index at which the recurrence begins.
- Write the recurrence clearly.
- Use already-known terms to calculate the next required term.
- Proceed in index order so that every dependency is available.
- Check that each computed term follows the stated recurrence.
A Reliable Procedure for Comparing the Two Forms
- Generate several terms from each description.
- Verify that the same indices produce the same values.
- Check that the same starting index is being used.
- Check that the initial value agrees with the explicit formula.
- Check that the explicit formula satisfies the recursive step.
Matching a few terms is useful evidence, but algebraic verification provides the stronger check when a general equivalence is being claimed.
Example: Verify an Explicit Formula against a Recurrence
Consider
\[ a_1=4,\qquad a_n=a_{n-1}+3. \]We claim that
\[ a_n=3n+1. \]First check the initial value:
\[ a_1=3(1)+1=4. \]Next compute the difference between consecutive explicit terms:
\[ a_n-a_{n-1} =(3n+1)-[3(n-1)+1]. \]Simplifying,
\[ a_n-a_{n-1}=3. \]Therefore,
\[ a_n=a_{n-1}+3, \]so the explicit formula matches both the initial value and the recursive step.
When Explicit Form Is Especially Useful
Explicit form is often useful when we want to:
- find a distant term quickly;
- compare term values at different positions;
- analyze how \(a_n\) depends directly on \(n\);
- graph the sequence using points \((n,a_n)\);
- study long-run behavior using a direct formula.
When Recursive Form Is Especially Useful
Recursive form is often useful when we want to:
- describe a repeated step or state transition;
- generate terms sequentially;
- model systems whose next state depends on previous states;
- encode algorithms and iterative procedures;
- describe sequences whose direct formulas are difficult to obtain.
Practice 1: Evaluate an Explicit Rule
Let
\[ a_n=4n-3. \]Find \(a_1\), \(a_5\), and \(a_{20}\).
Practice 2: List Terms from an Explicit Rule
Let
\[ b_n=n^2-1,\qquad n\ge 1. \]List the first five terms.
Practice 3: Start at Index Zero
Let
\[ c_n=3\cdot2^n,\qquad n\ge 0. \]List \(c_0,c_1,c_2,c_3\).
Practice 4: Generate a Recursive Sequence
Let
\[ a_1=6, \] \[ a_n=a_{n-1}+5\qquad(n\ge 2). \]List the first six terms.
Practice 5: Recursive Multiplication
Let
\[ b_1=2, \] \[ b_n=3b_{n-1}\qquad(n\ge 2). \]Find \(b_2,b_3,b_4\).
Practice 6: Two Previous Terms
Let
\[ c_1=2,\qquad c_2=3, \] \[ c_n=c_{n-1}+c_{n-2}\qquad(n\ge 3). \]Find \(c_3,c_4,c_5,c_6\).
Practice 7: Convert to Recursive Form
The sequence is defined explicitly by
\[ a_n=5n+2,\qquad n\ge 1. \]Write an equivalent recursive definition.
Practice 8: Convert a Constant-Step Recurrence to Explicit Form
The sequence is defined by
\[ a_1=9, \] \[ a_n=a_{n-1}+4\qquad(n\ge 2). \]Write an equivalent explicit formula.
Practice 9: Identify Missing Information
A student writes
\[ a_n=2a_{n-1} \]and claims that this uniquely defines a sequence.
Explain what information is missing.
Practice 10: Compare Representations
Consider
\[ a_n=3\cdot2^{n-1} \]and
\[ b_1=3,\qquad b_n=2b_{n-1}. \]Show that the first five terms agree.
Practice Answers
Practice 1
\[ a_1=4(1)-3=1, \] \[ a_5=4(5)-3=17, \] \[ a_{20}=4(20)-3=77. \]Therefore,
\[ \boxed{a_1=1,\qquad a_5=17,\qquad a_{20}=77}. \]Practice 2
\[ b_1=1^2-1=0, \] \[ b_2=2^2-1=3, \] \[ b_3=3^2-1=8, \] \[ b_4=4^2-1=15, \] \[ b_5=5^2-1=24. \]The first five terms are
\[ \boxed{0,\ 3,\ 8,\ 15,\ 24}. \]Practice 3
\[ c_0=3\cdot2^0=3, \] \[ c_1=3\cdot2^1=6, \] \[ c_2=3\cdot2^2=12, \] \[ c_3=3\cdot2^3=24. \]Therefore,
\[ \boxed{3,\ 6,\ 12,\ 24}. \]Practice 4
\[ a_1=6, \] \[ a_2=11,\qquad a_3=16,\qquad a_4=21,\qquad a_5=26,\qquad a_6=31. \]Thus,
\[ \boxed{6,\ 11,\ 16,\ 21,\ 26,\ 31}. \]Practice 5
\[ b_2=3(2)=6, \] \[ b_3=3(6)=18, \] \[ b_4=3(18)=54. \]Therefore,
\[ \boxed{b_2=6,\qquad b_3=18,\qquad b_4=54}. \]Practice 6
\[ c_3=3+2=5, \] \[ c_4=5+3=8, \] \[ c_5=8+5=13, \] \[ c_6=13+8=21. \]Therefore,
\[ \boxed{c_3=5,\qquad c_4=8,\qquad c_5=13,\qquad c_6=21}. \]Practice 7
The first term is
\[ a_1=5(1)+2=7. \]Consecutive terms differ by \(5\), so an equivalent recursive definition is
\[ \boxed{a_1=7,\qquad a_n=a_{n-1}+5\quad(n\ge 2)}. \]Practice 8
Starting from \(9\), the sequence adds \(4\) a total of \(n-1\) times to reach the \(n\)th term:
\[ a_n=9+4(n-1). \]Therefore,
\[ \boxed{a_n=4n+5}. \]Practice 9
The recurrence tells how to obtain each new term from the preceding term, but it does not give a starting value.
At least one initial value, such as \(a_1\), must be supplied to determine a unique sequence.
Practice 10
From the explicit rule,
\[ a_1=3,\quad a_2=6,\quad a_3=12,\quad a_4=24,\quad a_5=48. \]From the recursive rule,
\[ b_1=3,\quad b_2=6,\quad b_3=12,\quad b_4=24,\quad b_5=48. \]Therefore, the first five terms agree:
\[ \boxed{3,\ 6,\ 12,\ 24,\ 48}. \]What to Remember
A sequence is an ordered list of terms indexed by position.
An explicit rule gives a term directly:
\[ \boxed{a_n=f(n)}. \]A recursive rule gives later terms from earlier terms:
\[ \boxed{a_n=\text{expression involving earlier terms}}. \]- Check where the indexing begins.
- Keep the index separate from the term value.
- Recursive rules require sufficient initial values.
- Explicit rules often make distant terms easier to evaluate.
- Recursive rules often describe repeated processes more naturally.
- The same sequence may have both explicit and recursive descriptions.
- A finite list of terms does not, by itself, determine a unique rule.
Where We Go Next
Recursive descriptions depend on the values that start the process. Those starting values are not merely bookkeeping; they are part of what determines the sequence.
The next lesson, Initial Conditions, examines how initial values anchor recursive definitions, how many starting values a recurrence requires, and how changing those conditions can produce different sequences from the same recurrence rule.