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Arithmetic and Geometric Sequences

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Arithmetic and Geometric Sequences - Complete Interactive Lesson

Part 1: Sequences as Functions

🔢 Arithmetic & Geometric Sequences

Part 1 of 7 — Sequences as Functions

Topics in This Part

Section
What Is a Sequence?
Terms, Index, and Notation
Two Patterns That Rule Them All

🔑 Key Concept: A sequence is just a function whose inputs are the counting numbers $1, 2, 3, \dots$ and whose outputs are the terms $a_1, a_2, a_3, \dots$ . Everything in AP Precalculus about sequences comes from asking "how does each term grow from the one before it?"

What Is a Sequence?

A sequence is an ordered list of numbers:

$2,\; 5,\; 8,\; 11,\; 14,\; \dots$

Each number is a term. We label terms by their position $n$ (the index):

Concept Check — Reading Notation 🎯

Two Patterns That Rule Them All

Almost every sequence in AP Precalculus is built one of two ways. Look at how you get from each term to the next:

Sequence	Step from term to term	Name
$2, 5, 8, 11, \dots$	add $3$ each time	arithmetic
$3, 6, 12, 24$

Classify Each Sequence 🔽

Decide whether you add the same amount (arithmetic) or multiply by the same amount (geometric).

Measure the Step 🧮

For an arithmetic sequence, report the common difference $d = a_2 - a_1$ . For a geometric one, report the common ratio $r = \frac{a_{2}}{}$ .

Where We're Headed

You can now spot the two families. The rest of the lesson gives each family its own toolkit:

Part 2–3: Formulas for the $n$ -th term of arithmetic and geometric sequences.
Part 4: Recursive vs. explicit rules — two languages for the same pattern.
Part 5–6: Adding the terms up — series and the famous infinite sum.
Part 7: Mixed mastery and an exit quiz.

💡 Keep the mantra: add → arithmetic → linear, multiply → geometric → exponential.

Part 2: Arithmetic Sequences

🔢 Arithmetic & Geometric Sequences

Part 2 of 7 — Arithmetic Sequences

🔑 The Idea: An arithmetic sequence has a constant common difference $d$ . To find any term without listing them all, you start at $a_1$ and add $d$ the right number of times.

The $n$ -th Term Formula

Part 3: Geometric Sequences

🔢 Arithmetic & Geometric Sequences

Part 3 of 7 — Geometric Sequences

🔑 The Idea: A geometric sequence has a constant common ratio $r$ . Instead of adding $d$ , you multiply by $r$ — so the terms grow (or shrink) by the same factor each step.

The $n$ -th Term Formula

Start at $a_{}$ and multiply by each step. Reaching takes multiplications:

Part 4: Recursive vs. Explicit Rules

🔢 Arithmetic & Geometric Sequences

Part 4 of 7 — Recursive vs. Explicit Rules

🔑 Two languages, one sequence. An explicit rule jumps straight to any term from $n$ . A recursive rule defines each term using the previous one. Both describe the same list — you should be able to translate between them.

Explicit vs. Recursive

Explicit: a formula in terms of $n$ alone. Plug in $n$ , get the term.

$Arithmetic: a_{n} = a_{1} + (n - 1) d Geometric:$

Part 5: Series: Adding the Terms Up

🔢 Arithmetic & Geometric Sequences

Part 5 of 7 — Series: Adding the Terms Up

🔑 A series is a sum of a sequence's terms. Listing and adding by hand is hopeless for many terms — so each family has a closed-form sum formula that does it in one step.

Arithmetic Series

The sum of the first $n$ terms of an arithmetic sequence is the average of the first and last term, times the number of terms:

$S_n = \frac{n}{2}\,(a_1 + a_n)$

Part 6: Infinite Geometric Series

🔢 Arithmetic & Geometric Sequences

Part 6 of 7 — Infinite Geometric Series

🔑 The surprise: Adding infinitely many positive numbers can still give a finite total — but only for a geometric series whose terms shrink fast enough toward $0$ .

When Does an Infinite Sum Settle Down?

For a geometric series $a_1 + a_1 r + a_1 r^2 + \dots$ , look back at the finite formula .

Part 7: Applications, Mastery & Exit Quiz

🔢 Arithmetic & Geometric Sequences

Part 7 of 7 — Applications, Mastery & Exit Quiz

You now own both families end-to-end: classify, find any term, switch between recursive and explicit, and sum (finite or infinite). Let's apply it and then test it.

Quick Reference

Goal	Arithmetic (add $d$ )	Geometric (multiply by $r$ )
Find the step	$d = a_2 - a_1$

So $a_1 = 2$ , $a_3 = 8$ , and $a_5 = 14$ . The subscript is the input; the value is the output.

💡 AP framing: Because the inputs are the integers $1, 2, 3, \dots$ , a sequence is a function $a$ with $a(n) = a_n$ . This is why sequences behave like the discrete cousins of the lines and exponentials you already know.

3, 6, 12, 24, \dots

Arithmetic = constant difference (you add the same number). These grow linearly.
Geometric = constant ratio (you multiply by the same number). These grow exponentially.

🔑 The one question: To classify any sequence, ask "do I add the same thing, or multiply by the same thing?" Add → arithmetic. Multiply → geometric. Neither → it is some other kind of sequence.

r = \frac{a _{2}}{a _{1}}

1) Arithmetic $4,\; 11,\; 18,\; 25,\; \dots$ — find $d$ . 2) Geometric $7,\; 14,\; 28,\; 56,\; \dots$ — find $r$ . 3) Geometric $80,\; 40,\; 20,\; 10,\; \dots$ — find $r$ . (decimal or fraction is fine)

If you start at $a_1$ and add $d$ each step, then to reach term $a_n$ you take $(n-1)$ steps:

$a_n = a_1 + (n-1)\,d$

The common difference is $d = a_2 - a_1$ (any term minus the one before it).

Worked Example: $3,\; 7,\; 11,\; 15,\; \dots$

Here $a_1 = 3$ and $d = 7 - 3 = 4$ . The rule is:

$a_n = 3 + (n-1)(4) = 4n - 1$

$a_{10} = 3 + (10-1)(4) = 3 + 36 = 39$

⚠️ Watch the $(n-1)$ . It is $n-1$ , not $n$ , because the first term takes zero steps. Using $n$ is the single most common arithmetic-sequence error.

Why Arithmetic = Linear

Distribute the formula:

$a_n = a_1 + (n-1)d = \underbrace{d}_{\text{slope}}\cdot n + \underbrace{(a_1 - d)}_{\text{intercept}}$

This is $y = mx + b$ in disguise. The common difference $d$ is the slope — the sequence changes by the same amount over each equal step in $n$ .

Arithmetic sequence	Linear function
common difference $d$	slope $m$
add $d$ each step	rise $m$ per run of $1$

💡 AP connection: A sequence is arithmetic exactly when it changes by a constant amount over equal-length input intervals — the defining feature of a linear function.

Concept Check 🎯

Find the Term 🧮

Use $a_n = a_1 + (n-1)d$ .

1) $a_1 = 4,\; d = 6$ . Find $a_8$ . 2) $-5,\; -2,\; 1,\; 4,\; \dots$ Find $a_{12}$ . 3) $a_1 = 50,\; d = -7$ . Find $a_6$ .

Work Backwards 🧮

Sometimes you are handed two non-adjacent terms.

1) An arithmetic sequence has $a_1 = 7$ and $a_5 = 19$ . Find $d$ . (Hint: $a_5 = a_1 + 4d$ .) 2) Using that same sequence, find $a_{10}$ .

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

The common ratio is $r = \dfrac{a_2}{a_1}$ (any term divided by the one before it).

Worked Example: $5,\; 15,\; 45,\; 135,\; \dots$

Here $a_1 = 5$ and $r = \dfrac{15}{5} = 3$ . So:

$a_n = 5 \cdot 3^{\,n-1}$

$a_4 = 5 \cdot 3^{4-1} = 5 \cdot 27 = 135$

⚠️ The exponent is $n-1$ , not $n$ . Same reason as arithmetic: the first term has had zero multiplications applied.

Why Geometric = Exponential

The formula $a_n = a_1 \cdot r^{\,n-1}$ is an exponential function of $n$ — a constant base $r$ raised to a power.

Geometric sequence	Exponential function
common ratio $r$	base $b$
multiply by $r$ each step	multiply by $b$ per unit
$a_{n} = a_{1} \cdot r^{n - 1}$

The ratio $r$ controls the behavior:

$r > 1$ : terms grow (e.g. $r=2$ : $3,6,12,24,\dots$ )
: terms toward (e.g. : )

💡 AP connection: A sequence is geometric exactly when it changes by a constant ratio over equal-length input intervals — the defining feature of an exponential function.

Concept Check 🎯

Find the Term 🧮

Use $a_n = a_1 \cdot r^{\,n-1}$ . Decimals are fine.

1) $a_1 = 3,\; r = 2$ . Find $a_6$ . Find . . Find .

Find the Ratio (and a Term) 🔽

A geometric sequence has $a_1 = 2$ and $a_4 = 54$ .

Arithmetic: a_{n} = a_{1} + (n - 1) d Geometric: a_{n} = a_{1} \cdot r^{n - 1}

Recursive: a starting term plus a rule for getting the next term from the current one. A recursive definition always needs the seed $a_1$ .

$\text{Arithmetic: } a_1 = \text{(given)},\;\; a_n = a_{n-1} + d$ $\text{Geometric: } a_1 = \text{(given)},\;\; a_n = a_{n-1} \cdot r$

	Explicit	Recursive
Strength	jump to $a_{100}$ instantly	mirrors how the pattern builds
Needs	$a_1$ and $d$ or $r$	$a_1$ and the previous term

⚠️ A recursive rule with no starting value is incomplete — you cannot generate a single term without the seed $a_1$ .

Translating Between Them

Example: Recursive → terms

$a_1 = 2, \qquad a_n = a_{n-1} + 5$

This adds $5$ each step, so it is arithmetic with $d = 5$ :

$2,\; 7,\; 12,\; 17,\; \dots \qquad a_4 = 2 + 3(5) = 17$

Example: Recursive → terms (geometric)

$a_1 = 3, \qquad a_n = 2\,a_{n-1}$

This multiplies by $2$ each step, so it is geometric with $r = 2$ :

$3,\; 6,\; 12,\; 24,\; \dots \qquad a_4 = 3 \cdot 2^{3} = 24$

💡 Spotting the type from a recursive rule: if the rule adds a constant, it is arithmetic; if it multiplies by a constant, it is geometric.

Match the Rule to the Type 🔽

Concept Check 🎯

Generate and Convert 🧮

1) $a_1 = -1$ , $a_n = a_{n-1} + 6$ . Find $a_5$ . 2) $a_1 = 64$ , $a_n = \tfrac{1}{2}\,a_{n-1}$ . Find $a_4$ . 3) The explicit rule $a_n = 7 + (n-1)(4)$ has common difference $d = \,?$

If you do not yet know $a_n$ , substitute $a_n = a_1 + (n-1)d$ to get the equivalent form:

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

Worked Example: $3 + 7 + 11 + \dots$ (first $20$ terms)

Here $a_1 = 3$ , $d = 4$ , $n = 20$ :

$S_{20} = \frac{20}{2}\bigl(2(3) + (20-1)(4)\bigr) = 10\,(6 + 76) = 10 \cdot 82 = 820$

💡 Why the average works: pair the first term with the last, the second with the second-to-last — each pair sums to the same total $a_1 + a_n$ . There are $\frac{n}{2}$ such pairs.

Geometric Series (finite)

The sum of the first $n$ terms of a geometric sequence is:

$S_n = a_1 \cdot \frac{1 - r^{\,n}}{1 - r} \qquad (r \ne 1)$

Worked Example: $4 + 12 + 36 + \dots$ (first $5$ terms)

Here $a_1 = 4$ , $r = 3$ , $n = 5$ :

$S_5 = 4 \cdot \frac{1 - 3^{5}}{1 - 3} = 4 \cdot \frac{1 - 243}{-2} = 4 \cdot \frac{-242}{-2} = 4 \cdot 121 = 484$

⚠️ Use the right formula for the right family. Arithmetic uses $\frac{n}{2}(a_1 + a_n)$ ; geometric uses . Mixing them up is a classic exam slip.

Concept Check 🎯

Sum an Arithmetic Series 🧮

Use $S_n = \dfrac{n}{2}\bigl(2a_1 + (n-1)d\bigr)$ .

1) $a_1 = -5,\; d = 3$ . Find $S_{15}$ . . Find .

Sum a Geometric Series 🧮

Use $S_n = a_1 \dfrac{1 - r^{\,n}}{1 - r}$ .

1) $a_1 = 6,\; r = 2,\; n = 4$ . Find $S_4$ . . Find .

S_n = a_1\dfrac{1 - r^n}{1 - r}

If $|r| < 1$ , then $r^n \to 0$ as $n \to \infty$ , so $S_n$ approaches a fixed number — the series converges.
If $|r| \ge 1$ , the terms do not shrink to $0$ , the partial sums run off to infinity (or oscillate) — the series diverges.

When $|r| < 1$ , dropping the vanishing $r^n$ gives the infinite-sum formula:

$S_\infty = \frac{a_1}{1 - r} \qquad \text{(only when } |r| < 1\text{)}$

⚠️ The condition $|r| < 1$ is not optional. If $|r| \ge 1$ there is no finite sum, and plugging into the formula gives a meaningless answer.

Worked Examples

Example: $8 + 4 + 2 + 1 + \dots$

Here $a_1 = 8$ and $r = \dfrac{1}{2}$ . Since $|r| < 1$ :

$S_\infty = \frac{8}{1 - \frac{1}{2}} = \frac{8}{\frac{1}{2}} = 16$

Example: $12 + 4 + \tfrac{4}{3} + \dots$

Here $a_1 = 12$ and $r = \dfrac{1}{3}$ :

$S_\infty = \frac{12}{1 - \frac{1}{3}} = \frac{12}{\frac{2}{3}} = 12 \cdot \frac{3}{2} = 18$

💡 Sanity check: the total must be bigger than $a_1$ (you keep adding positive terms) but not by an unbounded amount. $16 > 8$ ✓ and $18 > 12$ ✓.

Concept Check — Converge or Diverge? 🎯

Build the Infinite Sum 🔽

Evaluate $S_\infty$ for $3 + 2 + \tfrac{4}{3} + \dots$

Infinite Sums 🧮

Use $S_\infty = \dfrac{a_1}{1 - r}$ (only valid when $|r| < 1$ ).

1) $a_1 = 10,\; r = \frac{1}{2}$ . Find . . Find .

⚠️ Three traps: use $n-1$ (not $n$ ) in the term formulas; pick the sum formula that matches the family; and only use $S_\infty$ when $|r| < 1$ .

Modeling With Sequences

Arithmetic — a raise of a fixed dollar amount. A job pays $40,000 in year $1$ and adds a flat $2,500 raise each year. Salary is arithmetic with $a_1 = 40000$ , $d = 2500$ :

$a_{10} = 40000 + (10-1)(2500) = 40000 + 22500 = 62500$

Total earned over $10$ years (an arithmetic series):

$S_{10} = \frac{10}{2}(a_1 + a_{10}) = 5\,(40000 + 62500) = 5 \cdot 102500 = 512500$

Geometric — a quantity that changes by a fixed percent. A population multiplies by $r = 1.05$ ( $5\%$ growth) each year. After $3$ years of growth from $1000$ :

$a_4 = 1000 \cdot (1.05)^{3} \approx 1157.63$

💡 Decide the family from the wording: "increases by a fixed amount" → arithmetic; "increases by a fixed percent/factor" → geometric.

Mixed Practice 🎯

Choose the Right Tool 🔽

For each task, pick the formula you'd use.

Exit Quiz ✅

Answer all three to finish the lesson.

a_{n} = a_{1} + (n - 1) d

a_{n} = a_{1} \cdot r^{n - 1}

r < 0

: terms alternate sign (e.g.

r=-2

1,-2,4,-8,\dots

)

2,\; 1,\; 0.5,\; 0.25,\; \dots

a_1 = 100,\; r = 0.8

a_1\frac{1-r^n}{1-r}

a_1 = 4,\; r = 3,\; n = 5

a_1 = 18,\; r = \frac{1}{3}

Arithmetic and Geometric Sequences

Arithmetic and Geometric Sequences - Complete Interactive Lesson

Part 1: Sequences as Functions

🔢 Arithmetic & Geometric Sequences

Topics in This Part

What Is a Sequence?

Two Patterns That Rule Them All

Where We're Headed

Part 2: Arithmetic Sequences

🔢 Arithmetic & Geometric Sequences

The nn-th Term Formula

Part 3: Geometric Sequences

🔢 Arithmetic & Geometric Sequences

The nnn-th Term Formula

Part 4: Recursive vs. Explicit Rules

🔢 Arithmetic & Geometric Sequences

Explicit vs. Recursive

Part 5: Series: Adding the Terms Up

🔢 Arithmetic & Geometric Sequences

Arithmetic Series

Part 6: Infinite Geometric Series

🔢 Arithmetic & Geometric Sequences

When Does an Infinite Sum Settle Down?

Part 7: Applications, Mastery & Exit Quiz

🔢 Arithmetic & Geometric Sequences

Quick Reference

Worked Example: 3, 7, 11, 15, …3,\; 7,\; 11,\; 15,\; \dots3,7,11,15,…

Why Arithmetic = Linear

Worked Example: 5, 15, 45, 135, …5,\; 15,\; 45,\; 135,\; \dots5,15,45,135,…

Why Geometric = Exponential

Translating Between Them

Example: Recursive → terms

Example: Recursive → terms (geometric)

Worked Example: 3+7+11+…3 + 7 + 11 + \dots3+7+11+… (first 202020 terms)

Geometric Series (finite)

Worked Example: 4+12+36+…4 + 12 + 36 + \dots4+12+36+… (first 555 terms)

Worked Examples

Example: 8+4+2+1+…8 + 4 + 2 + 1 + \dots8+4+2+1+…

Example: 12+4+43+…12 + 4 + \tfrac{4}{3} + \dots12+4+34​+…

Modeling With Sequences

The $n$ -th Term Formula

The $n$ -th Term Formula

Worked Example: $3,\; 7,\; 11,\; 15,\; \dots$

Worked Example: $5,\; 15,\; 45,\; 135,\; \dots$

Worked Example: $3 + 7 + 11 + \dots$ (first $20$ terms)

Worked Example: $4 + 12 + 36 + \dots$ (first $5$ terms)

Example: $8 + 4 + 2 + 1 + \dots$

Example: $12 + 4 + \tfrac{4}{3} + \dots$