Presents a sequence as a function whose domain is the counting numbers, introduces subscript notation, and distinguishes explicit formulas from recursive ones — including why a recursive definition needs a starting value. Covers factorial notation and the conventions attached to it.
Subject: Precalculus · 65 slides · symbolic lesson
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Title
Precalculus · Chapter 11 — Sequences, Probability and Counting Theory
§11.1 Sequences and Their Notations, pp. 1290-1306
Objectives
Five things, each one you can check yourself on paper.
OpenStax, Precalculus, §11.1 Sequences and Their Notations §11.1, pp. 1290-1306 — the pages these objectives are drawn from
Warm-up
Chapter 1 gave a general framework. This fits inside it.
Discussion prompt
The list 3, 6, 9, 12 assigns a number to each position. What kind of object is that?
Hint: One input, one output.
Answer:
Each position gets exactly one number. That is precisely the definition of a function.
The inputs are the counting numbers — first, second, third — rather than every real number.
So a sequence is a function with a restricted domain. Everything from chapter 1 applies, and the only new thing is notation for recording that restriction.
Concept
A sequence assigns one number to each counting number. Because the domain is discrete, its graph is a set of separated dots rather than a curve.
sequence — a function whose domain is the counting numbers, so that each term has a position in a list
\[ a_n=3n: \quad 3,\;6,\;9,\;12,\;\ldots \]
The subscript takes the place of parentheses and signals the restricted domain. Beyond that, a sequence behaves like any other function — it can be increasing, bounded, or have a limiting behaviour, all of which chapter 1's vocabulary already covers.
Figure (svg): A diagram showing a sequence as a function whose inputs are the counting numbers and whose outputs are the terms
OpenStax, Precalculus, §11.1 Sequences and Their Notations §11.1, pp. 1290-1294
Section
Section 1
Concept
Everything chapter 1 said about functions applies, with the domain cut down to the counting numbers and the notation changed to record it.
Some sequences are indexed from zero rather than one, which is a choice the problem makes rather than a rule. Checking which index the first term carries is worth doing before computing anything, since an off-by-one error propagates through everything after.
Figure (svg): A contrast between ordinary function notation and sequence notation for the same idea
OpenStax, Precalculus, §11.1 Sequences and Their Notations §11.1, pp. 1290-1295
Picture it
The subscript is the only real difference.
Figure (svg): A contrast between ordinary function notation and sequence notation for the same idea
The third row on each side is the visible consequence. A restricted domain means the graph has gaps, which is why sequence graphs are plotted as isolated points.
Worked example
Substitute the first few indices.
\[ \text{List the first four terms of } a_n=2n+1. \]
Substitute the first index
Why: One.
\[ 3 \]
Substitute the second
Why: Two.
\[ 5 \]
Substitute the third
Why: Three.
\[ 7 \]
Substitute the fourth
Why: Four.
\[ 9 \]
Figure (svg): A diagram showing a sequence as a function whose inputs are the counting numbers and whose outputs are the terms
\[ 3,\;5,\;7,\;9 \]
Verify: check the pattern
Why: Each term exceeds the previous by 2, which matches the coefficient of the index in the formula. That consistency is a quick check that no substitution went wrong.
OpenStax, Precalculus, §11.1 Sequences and Their Notations §11.1, pp. 1291-1293
Prediction
A sequence is plotted with position horizontally and value vertically.
Predict first
What does the graph look like?
Correct: Separated dots.
Why: The domain is the counting numbers, so the function has values only at those inputs and nothing in between. That discreteness is the visible consequence of the restricted domain.
Worked example
Solve rather than substitute.
\[ \text{Which term of } a_n=2n+1 \text{ equals } 41? \]
Set the formula equal
Why: To the target value.
\[ 2 n + 1 = 41 \]
Solve for the index
Why: Ordinary algebra.
\[ n = 20 \]
Check it is a counting number
Why: It is.
State the answer
Why: The twentieth term.
\[ n = 20 \]
Figure (svg): The solution to Worked example find a term's position shown as a ladder of expressions, one row per legal move
\[ n=20 \]
Verify: check that a non-integer answer would be meaningful
Why: Had the equation given a fractional index, the value would simply not appear in the sequence — since positions are counting numbers. That check matters whenever a target value is not obviously in the list.
OpenStax, Precalculus, §11.1 Sequences and Their Notations §11.1, pp. 1293-1295
Trap
\[ \text{the first term is }a_1\text{, always} \]
Take the indexing for granted
Why: The problem's own starting index is not checked.
A sequence indexed from zero has every term off by one position.
Check where the indexing starts. Many sequences begin at one, but some begin at zero.
The choice is made by the problem, and the formula is written to suit it.
An off-by-one error propagates through everything after, so the check is worth the two seconds it costs.
Faded example
From an explicit formula at the fourth position.
Fill in the blanks
a_4=2(4)+1=9
Why: Substituting the index into the formula gives the term directly. An explicit formula makes any term computable in one step, whatever its position.
Sorting
The domain must be the counting numbers.
Sort into buckets
Sort each object.
Explain it to yourself
Chapter 1 used parentheses and this uses subscripts.
Discussion prompt
Explain what the subscript signals.
Hint: What is different about the input?
Answer:
The subscript signals that the input is a position — a counting number — rather than any value at all.
It is a reminder built into the notation, so that reading an expression tells you the domain is restricted without having to state it separately.
Nothing else changes: every idea from chapter 1 still applies. A good explanation notes that this is why the section can move quickly to recursion, which is the one genuinely new idea here.
Section
Section 2
Concept
An explicit formula gives each term directly from its position, so the hundredth term takes no more work than the first.
The last point is a real caution. A finite list never determines a formula uniquely — infinitely many rules agree on any given first few terms and then diverge. Finding a pattern is a reasonable guess, not a deduction.
Figure (svg): A contrast between an explicit formula, which computes any term directly, and a recursive one, which computes each term from its predecessor
OpenStax, Precalculus, §11.1 Sequences and Their Notations §11.1, pp. 1295-1299
Picture it
The left column is this idea.
Figure (svg): A contrast between an explicit formula, which computes any term directly, and a recursive one, which computes each term from its predecessor
The third row is the practical difference. An explicit formula reaches the hundredth term immediately where a recursive one requires everything before it.
Worked example
Look at how the terms change.
\[ \text{Find a formula for } 5,\;8,\;11,\;14,\;\ldots \]
Find the differences
Why: Between consecutive terms.
\[ 3\text{ each time} \]
Note the pattern
Why: A constant difference means linear.
\[ 3 n\text{ plus something} \]
Fit the first term
Why: Three times one is three, but the term is five.
\[ \text{add } 2 \]
Write the formula
Why: And check.
\[ 3 n + 2 \]
Figure (svg): A contrast between an explicit formula, which computes any term directly, and a recursive one, which computes each term from its predecessor
\[ a_n=3n+2 \]
Verify: check a later term
Why: The fourth term should be 12 plus 2, which is 14 — matching the given list. Checking a term other than the first is what confirms the formula rather than just the fitting.
OpenStax, Precalculus, §11.1 Sequences and Their Notations §11.1, pp. 1296-1298
Prediction
The differences between consecutive terms are constant.
Predict first
What kind of formula fits?
Correct: Linear in the index.
Why: A constant difference means each step adds the same amount, which is exactly what a linear formula does. The constant difference becomes the coefficient of the index.
Worked example
A power of negative one does the work.
\[ \text{Find a formula for } -2,\;4,\;-6,\;8,\;\ldots \]
Ignore the signs
Why: The sizes.
\[ 2, 4, 6, 8 \]
Find that pattern
Why: Twice the index.
\[ 2 n \]
Handle the alternation
Why: A power of negative one.
\[ (-1) ^{n} \]
Fit the first sign
Why: The first term is negative.
\[ (-1) ^{n} \times 2 n \]
Figure (svg): The solution to Worked example alternating signs shown as a ladder of expressions, one row per legal move
\[ a_n=(-1)^n\cdot 2n \]
Verify: check two consecutive terms
Why: At the first position the power is negative one, giving negative 2 — correct. At the second it is positive one, giving 4 — also correct. Checking two consecutive terms is what confirms the alternation starts on the right foot.
OpenStax, Precalculus, §11.1 Sequences and Their Notations §11.1, pp. 1298-1299
Error analysis
A student finds a formula from four terms.
Annotate
On: \( 1,\;2,\;4,\;8 \;\Longrightarrow\; a_n=2^{n-1}, \text{ so the fifth term is }16 \)
In practice the simplest fitting formula is the intended answer, and that convention is what makes such questions well posed. But it is a convention about intent rather than a mathematical certainty.
Faded example
Differences of 3, first term 5.
Fill in the blanks
a_n=3n+2, \text23(1)+___=5
Why: The common difference gives the coefficient and the first term fixes the constant. Checking a later term confirms the fit rather than just the construction.
Sorting
Different features need different pieces.
Sort into buckets
Sort each feature.
Explain it
Four terms do not determine a formula.
Discussion prompt
Explain to a classmate why, and what the convention is.
Hint: How many rules fit?
Answer:
Infinitely many formulas agree on any finite list of terms and then diverge. Nothing in four numbers rules out a fifth that breaks the pattern.
So finding a formula is a reasonable guess rather than a deduction — an inference about what was intended.
The convention is that the simplest fitting formula is the intended one, which is what makes such questions well posed. A good explanation notes that this is a convention about communication, not a mathematical fact.
Section
Section 3
Concept
A recursive formula computes each term from the one before it. It defines nothing without a starting value, because the rule has nothing to build on.
The number of starting values must match how many previous terms the rule uses. A rule reaching back two terms needs two starting values, which is why the Fibonacci sequence specifies both its first and its second term.
Figure (svg): A contrast between an explicit formula, which computes any term directly, and a recursive one, which computes each term from its predecessor
OpenStax, Precalculus, §11.1 Sequences and Their Notations §11.1, pp. 1299-1303
Picture it
The right column is this idea.
Figure (svg): A contrast between an explicit formula, which computes any term directly, and a recursive one, which computes each term from its predecessor
The last row on the right is the requirement that is easiest to omit. A recursive rule on its own is an incomplete definition, not a shorthand one.
Worked example
Start from the given value.
\[ \text{List four terms of } a_n=2a_{n-1}+1, \; a_1=3. \]
Read the first term
Why: Given.
\[ 3 \]
Apply the rule
Why: Twice the previous plus one.
\[ 7 \]
Apply it again
Why: To the new term.
\[ 15 \]
Once more
Why: Continuing.
\[ 31 \]
Figure (svg): A contrast between an explicit formula, which computes any term directly, and a recursive one, which computes each term from its predecessor
\[ 3,\;7,\;15,\;31 \]
Verify: look for the pattern
Why: Each term is one less than a power of two: 4 minus 1, 8 minus 1, 16 minus 1, 32 minus 1. That suggests an explicit formula exists, which is often but not always the case for a recursive definition.
OpenStax, Precalculus, §11.1 Sequences and Their Notations §11.1, pp. 1300-1302
Prediction
A rule computes each term from the two before it.
Predict first
How many starting values are needed?
Correct: Two.
Why: The number of starting values must match how many previous terms the rule reaches back to. With one supplied, the first computed term would have a missing ingredient and the sequence could not start.
Worked example
Two starting values are needed.
\[ \text{List six terms of } a_n=a_{n-1}+a_{n-2}, \; a_1=1, \; a_2=1. \]
Read the two given terms
Why: Both one.
\[ 1, 1 \]
Add them
Why: For the third.
\[ 2 \]
Continue
Why: Each from the two before.
\[ 3, 5 \]
One more
Why: The sixth.
\[ 8 \]
Figure (svg): The solution to Worked example a rule reaching back two terms shown as a ladder of expressions, one row per legal move
\[ 1,\;1,\;2,\;3,\;5,\;8 \]
Verify: check why two starting values were needed
Why: The rule reaches back two positions, so computing the third term requires both the first and the second. With only one given, the third would be undefined and the whole sequence would fail to start.
OpenStax, Precalculus, §11.1 Sequences and Their Notations §11.1, pp. 1302-1303
Trap
\[ a_n=2a_{n-1}+1 \]
State the rule alone
Why: The starting value is treated as implied.
Every term depends on the one before, and nothing anchors the chain.
A recursive definition needs both a rule and a starting value. The rule alone describes infinitely many different sequences.
With a first term of 3 the sequence is one thing; with a first term of 0 it is another, and the rule cannot distinguish them.
Count how many previous terms the rule uses and supply that many starting values.
Faded example
With the previous term equal to 7.
Fill in the blanks
a_n=2(7)+1=15
Why: Each application uses the immediately preceding term, so the terms must be generated in order. Skipping ahead is impossible without computing everything in between.
Sorting
A rule needs the right number of starting values.
Sort into buckets
Sort each definition.
Step zero
You are given a recursive definition to work with.
Discussion prompt
What do you check before generating anything?
Hint: Count two things.
Answer:
How many previous terms the rule uses, and how many starting values are supplied. Those two numbers must match.
If the rule reaches back two positions and only one starting value is given, the definition is incomplete and no terms can be generated.
Then generate strictly in order, since each term depends on its predecessors. There is no way to jump ahead, which is the practical cost of a recursive definition.
Section
Section 4
Concept
An explicit formula reaches any term at once; a recursive one often describes the underlying process more naturally. Which is better depends on what is being asked.
Many sequences have both descriptions, and converting between them is a standard task. Some recursive sequences have no elementary explicit formula at all, which is when the recursive form is not merely preferable but necessary.
| you want to | prefer |
|---|---|
| find a distant term | explicit |
| describe a step-by-step process | recursive |
| model growth from a previous state | recursive |
| compute many terms at once | explicit |
| state a compound-interest rule | either, and both are used |
Figure (svg): A contrast between an explicit formula, which computes any term directly, and a recursive one, which computes each term from its predecessor
OpenStax, Precalculus, §11.1 Sequences and Their Notations §11.1, pp. 1303-1305
Picture it
Each column names one form's strengths.
Figure (svg): A contrast between an explicit formula, which computes any term directly, and a recursive one, which computes each term from its predecessor
Neither is better in general. The recursive form describes how a process works and the explicit form answers where it ends up.
Worked example
A constant increment becomes a multiple.
\[ \text{Convert } a_n=a_{n-1}+4, \; a_1=7 \text{ to an explicit formula.} \]
Note the increment
Why: Four each step.
Count the steps to term n
Why: One fewer than the index.
\[ n - 1\text{ steps} \]
Write the total added
Why: Four per step.
\[ 4(n - 1) \]
Add the first term
Why: The starting point.
\[ 7 + 4(n - 1) \]
Figure (svg): A contrast between an explicit formula, which computes any term directly, and a recursive one, which computes each term from its predecessor
\[ a_n=4n+3 \]
Verify: check both ends
Why: At the first position the formula gives 7, matching the starting value. At the second it gives 11, which is 7 plus 4 — matching the rule. Both the anchor and the increment are reproduced.
OpenStax, Precalculus, §11.1 Sequences and Their Notations §11.1, pp. 1304-1305
Prediction
You need the five-hundredth term of a sequence.
Predict first
Which form is better?
Correct: Explicit, since it reaches any term directly.
Why: A recursive formula would require computing all 499 terms before it. An explicit one substitutes the index once, which is the whole reason for converting when a distant term is wanted.
Worked example
Not every sequence has an elementary explicit formula.
\[ \text{Why is the Fibonacci sequence usually given recursively?} \]
State the rule
Why: Each term the sum of the two before.
Consider an explicit formula
Why: One exists but involves irrational powers.
Compare the descriptions
Why: The rule is far clearer.
Conclude
Why: The recursive form is the natural one.
Figure (svg): The solution to Worked example when recursion is necessary shown as a ladder of expressions, one row per legal move
\[ \text{recursion is the natural description} \]
Verify: consider what each form is good for
Why: The recursive rule generates terms quickly and describes the process transparently. The explicit formula would answer questions about the hundredth term directly, but at the cost of an expression involving irrational numbers that must combine to give a whole number every time.
OpenStax, Precalculus, §11.1 Sequences and Their Notations §11.1, pp. 1305-1306
Error analysis
A student converts a recursive formula with increment 4 and first term 7.
Annotate
On: \( a_n=7+4n \)
The number of steps to reach a term is one fewer than its index, because the first term takes no steps at all. Substituting the first index into any conversion is the standard check.
Faded example
Reaching the tenth term from the first.
Fill in the blanks
\text1=10-9=___
Why: The first term takes no steps, so reaching the tenth takes nine. That off-by-one is the standard error in converting a recursive formula to an explicit one.
Sorting
The question decides.
Sort into buckets
Sort each task.
Explain it
Both forms describe the same sequence.
Discussion prompt
Explain to a classmate when each is preferable.
Hint: What is each good at?
Answer:
An explicit formula gives any term in one substitution, so it wins whenever a distant term is wanted or many terms are needed at once.
A recursive formula describes how each step follows from the last, which often matches the underlying process — a population, an interest calculation, a physical iteration.
And some sequences have no simple explicit formula, so the recursive form is not merely preferable but the only practical description. A good explanation notes that converting between them is itself a standard skill, since the two answer different questions about the same object.
Section
Section 5
Concept
A factorial multiplies a counting number by every smaller one down to one. Zero factorial is defined to be one, which is a convention chosen to make later formulas work.
Simplifying a quotient of factorials is the usual computational task, and it almost never requires evaluating either one. Writing the larger factorial as a product down to the smaller lets most factors cancel.
Figure (svg): A card showing factorial notation as a descending product, with the convention that zero factorial is one
OpenStax, Precalculus, §11.1 Sequences and Their Notations §11.1, pp. 1305-1306
Picture it
A descending product, and the convention about zero.
Figure (svg): A card showing factorial notation as a descending product, with the convention that zero factorial is one
The bottom line is worth noticing now. Factorials outgrow exponentials, which is why they dominate the counting formulas where both appear.
Worked example
Cancel rather than evaluate.
\[ \text{Simplify } \frac{8!}{6!}. \]
Expand the larger
Why: Down to the smaller.
\[ 8 \times 7 \times 6! \]
Write the quotient
Why: The smaller factorial appears twice.
\[ 8 \times 7 \times 6!\text{ over } 6! \]
Cancel
Why: The common factorial.
\[ 8 \times 7 \]
Compute
Why: Multiply.
\[ 56 \]
Figure (svg): A card showing factorial notation as a descending product, with the convention that zero factorial is one
\[ 56 \]
Verify: check by evaluating
Why: Eight factorial is 40320 and six factorial is 720, and dividing gives 56 — matching. But the cancellation reached it with two multiplications rather than two large ones, which is why quotients are simplified rather than evaluated.
OpenStax, Precalculus, §11.1 Sequences and Their Notations §11.1, pp. 1305-1306
Prediction
By the standard convention.
Predict first
What is it?
Correct: One.
Why: It is defined to be one, which is the value that makes the counting formulas of the later sections come out correctly without special cases. Conventions of this kind are usually chosen for exactly that reason.
Worked example
The same cancellation, symbolically.
\[ \text{Simplify } \frac{(n+1)!}{(n-1)!}. \]
Expand the larger
Why: Down to the smaller.
\[ (n + 1) \times n \times(n - 1)! \]
Write the quotient
Why: The smaller appears in both.
Cancel
Why: The common factorial.
\[ (n + 1) \times n \]
Expand
Why: If wanted.
\[ n ^{2} + n \]
Figure (svg): The solution to Worked example simplify with a variable shown as a ladder of expressions, one row per legal move
\[ n(n+1) \]
Verify: test at a value
Why: At n equal to 5 the expression gives 30, and directly the quotient is 720 over 24, which is 30 — matching. Testing one value confirms the symbolic cancellation.
OpenStax, Precalculus, §11.1 Sequences and Their Notations §11.1, pp. 1306-1306
Trap
\[ \frac{8!}{6!}=\frac{8}{6}=\frac{4}{3} \]
Cancel the factorial symbols
Why: The exclamation marks are treated as removable.
The answer is 56, not four thirds.
A factorial is a product, not a factor. The symbols cannot be cancelled against each other.
Expanding the larger factorial down to the smaller creates a genuine common factor, which then cancels legitimately.
The expansion is the whole technique, and it works symbolically as well as numerically.
Faded example
Expanding down to the smaller factorial.
Fill in the blanks
\frac756=\frac______=8\cdot___=___
Why: Expanding the larger factorial only as far as the smaller creates a common factor that cancels. Neither factorial ever has to be evaluated.
Sorting
Factorials are products, not factors.
Sort into buckets
Sort each step.
Explain it to yourself
Zero factorial is defined to be one.
Discussion prompt
Explain why a convention was needed and why that value.
Hint: What would the descending product give?
Answer:
The descending product has nothing to multiply when the number is zero, so the definition does not extend by itself. A convention has to be chosen.
One is chosen because it makes the counting formulas of §11.5 and §11.6 come out correctly with no special case for zero — those formulas divide by factorials, and dividing by one leaves the count unchanged.
It is also consistent with the pattern: each factorial is the next one divided by that number, and applying that to one factorial gives one over one. A good explanation notes that conventions are usually chosen to remove special cases, which is exactly what happened here.
Comparison
Fill the blanks from memory. Both describe the same sequences differently.
Comparison matrix
| explicit | recursive | |
|---|---|---|
| computes a term from | its position | the previous term or terms |
| needs a starting value | no | yes, one per term reached back |
| reaching the hundredth term | one substitution | ninety-nine computations |
| describes a process | less naturally | directly |
The second row is the requirement most easily forgotten. A recursive rule without a starting value describes infinitely many sequences and specifies none.
Pattern
Five steps, and the first two prevent the commonest errors.
Step 5's two checks catch different things: the first catches an off-by-one in the indexing and the second catches an error in the rule itself.
OpenStax Algebra and Trigonometry 2e, §13.1 Sequences and Their Notations §13.1
Check
What a sequence is.
Check your understanding
A sequence is a function with what domain?
Answer: A
Why: Each position gets exactly one value, which is a function whose inputs are the counting numbers. That restricted domain is why the graph is separated dots rather than a curve.
Check
Recursive definitions.
Check your understanding
What does a recursive formula need besides its rule?
Answer: A
Why: Every term is defined from earlier ones, so something must anchor the chain. The number of starting values must match how many previous terms the rule reaches back to.
Check
Factorials.
Check your understanding
What is the value of zero factorial?
Answer: A
Why: It is defined to be one, chosen so that the counting formulas of the later sections work without a special case. The descending product gives no value by itself, so a convention was needed.
Real world
Recursion is how iterative computations are actually written.
Discussion prompt
A program computes compound interest year by year. Why is that a recursive description?
Hint: What does each year's balance depend on?
Answer:
Each year's balance is the previous year's multiplied by a growth factor. That is a recursive rule with the initial deposit as the starting value.
An explicit formula also exists — the deposit times the factor raised to the number of years — and it answers 'what will it be in twenty years' in one step.
Both are used, for different purposes: the recursive form models the process year by year and handles deposits or withdrawals along the way; the explicit form gives the endpoint immediately. A spreadsheet uses the first and a calculator the second, which is the same trade-off this section describes.
Commit first
State your confidence along with your answer.
Predict first
Why does a recursive formula need a starting value?
Correct: Every term is defined from earlier ones, so something must anchor the chain.
Why: The rule says how to get from one term to the next but never produces a first term. Without one supplied, the same rule describes infinitely many different sequences and specifies none of them.
Explain it
The test of understanding is being able to say why, not just what.
Discussion prompt
A classmate asks why sequences need their own notation when they are just functions.
Hint: What does the subscript record?
Answer:
They are just functions — with the domain restricted to the counting numbers. Nothing from chapter 1 stops applying.
The subscript records that restriction in the notation itself, so reading an expression tells you the input is a position rather than any number.
It is a signalling convention rather than a new system. A good explanation notes that the genuinely new idea in the section is recursion, which has no analogue in chapter 1 — defining a term from its predecessor rather than from its position.
Exit ticket
One honest answer, so the next lesson can start in the right place.
Predict first
Which idea from this lesson would you most want to see again?
Correct: Any of these is a legitimate answer; the useful one is the honest one.
Why: There is no correct choice here. The third is the genuinely new idea and the one most often left incomplete. The fourth becomes essential in §11.5 and §11.6, where factorials do all the counting.
Connect it up
One page, drawn from memory, is worth more than rereading the section.
Draw it
Write one sequence in both explicit and recursive form, marking the starting value on the recursive one. Beside it, note what each form is good at. Underneath, list the first six Fibonacci terms with the rule and both starting values, and simplify one quotient of factorials by cancelling rather than evaluating.
If your recursive form carries its starting value and your factorial quotient never evaluates either factorial, the section's two practical points are on the page.
Recap
Five things, and the third is the one to state completely.
| if you remember one thing | it should be this |
|---|---|
| about sequences | a function with the domain cut down to positions |
| about explicit formulas | any term in one step, but a pattern is a guess |
| about recursion | a rule without a starting value defines nothing |
| about factorials | expand down to the smaller one and cancel |
Section 11.2 takes the special case where each term differs from the last by a constant — the sequences whose explicit formulas are linear — and develops both their term and sum formulas.
OpenStax, Precalculus, §11.1 Sequences and Their Notations §11.1, pp. 1290-1306 — everything on these slides traces back here
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