Arithmetic and Geometric Sequences

🔢 Arithmetic & Geometric Sequences

Part 1 of 7 — Sequences as Functions

Topics in This Part

🔑 Key Concept: A sequence is just a function whose inputs are whole numbers (counting from $0$ or from $1$) 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):

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 whole numbers, a sequence is a function $a$ with $a(n) = a_n$, and its graph is a set of separate points. This is why sequences behave like the discrete cousins of the lines and exponentials you already know.

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:

• 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.

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 $r = \dfrac{a_2}{a_1}$.

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)

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: Rules from any known term, and how the two families grow differently.
• Part 6: Connecting sequences to linear and exponential functions.
• Part 7: Mixed mastery and an exit quiz.

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

🔢 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

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:

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

Find $a_{10}$:

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

⚠️ 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 $n$.

💡 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 $a_{10}$.

🔢 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 $a_1$ and multiply by $r$ each step. Reaching $a_n$ takes $(n-1)$ multiplications:

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

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:

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

Find $a_4$:

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

⚠️ 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.

The ratio $r$ controls the behavior:

• $r > 1$: terms grow (e.g. $r=2$: $3,6,12,24,\dots$)
• $0 < r < 1$: terms decay toward $0$ (e.g. $r=\frac{1}{2}$: $16,8,4,2,\dots$)
• $r < 0$: terms alternate sign (e.g. $r=-2$: $1,-2,4,-8,\dots$)

💡 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 $a_6$. 2) $2,\; 1,\; 0.5,\; 0.25,\; \dots$ Find $a_5$. 3) $a_1 = 100,\; r = 0.8$. Find $a_3$.

Find the Ratio (and a Term) 🔽

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

🔢 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.

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

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

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

⚠️ 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 $d = \,?$

🔢 Arithmetic & Geometric Sequences

Part 5 of 7 — Any-Term Rules & Comparing Growth

🔑 Two upgrades: You can write a rule starting from any known term, not just the first one. And you can predict how an arithmetic sequence and a geometric sequence grow differently, step by step.

Rules from Any Known Term

The AP Precalculus course often calls the initial value term $0$, so the basic rules are

$a_n = a_0 + dn \qquad\qquad g_n = g_0\, r^{\,n}$

If instead you know term $k$, count the steps $n - k$ from it:

$a_n = a_k + d\,(n - k) \qquad\qquad g_n = g_k\, r^{\,n - k}$

Arithmetic: $a_7 = 50$ and $d = 4$

• Rule: $a_n = 50 + 4(n - 7)$
• $a_{12} = 50 + 4(5) = 70$
• $a_0 = 50 + 4(-7) = 22$

Geometric: $g_3 = 40$ and $r = 0.5$

• Rule: $g_n = 40\,(0.5)^{\,n - 3}$
• $g_6 = 40\,(0.5)^3 = 5$
• $g_0 = 40\,(0.5)^{-3} = 40 \cdot 8 = 320$

💡 A negative step count $n - k$ just means you are walking backward from the known term.

Concept Check 🎯

Equal Steps vs. Growing Steps

Both sequences below start at $20$ and gain $10$ on the first step.

• The arithmetic sequence gains $10$, $10$, $10$, $10$: it increases equally with each step.
• The geometric sequence gains $10$, $15$, $22.5$, $33.75$: with positive terms and $r > 1$, it increases by a larger amount with each successive step.

⚠️ A decreasing geometric sequence of positive terms ($0 < r < 1$) drops by smaller and smaller amounts, while a decreasing arithmetic sequence drops by the same amount every step.

How Do They Grow? 🔽

Compare the Growth 🧮

Use $a_n = 20 + 10n$ and $g_n = 20(1.5)^n$.

1) $a_6$ 2) $g_4$ 3) How much $g$ gains from $n = 3$ to $n = 4$

🔢 Arithmetic & Geometric Sequences

Part 6 of 7 — From Sequences to Linear & Exponential Functions

🔑 Same structure, bigger domain. A linear function is an arithmetic sequence that accepts every real input; an exponential function is a geometric sequence that does the same. Both kinds of function can be built from a known point, just like the any-term rules.

Matching Forms

Building a Linear Function from Two Points: $(3, 14)$ and $(7, 30)$

$m = \frac{30 - 14}{7 - 3} = 4 \qquad f(x) = 14 + 4(x - 3) = 2 + 4x$

Building an Exponential Function from Two Points: $(0, 50)$ and $(2, 72)$

$b^{2} = \frac{72}{50} = 1.44 \;\Rightarrow\; b = 1.2 \qquad f(x) = 50\,(1.2)^{x}$

💡 Two distinct values always determine an arithmetic sequence, a geometric sequence, a linear function, or an exponential function.

Concept Check 🎯

Linear or Exponential? Check Equal Input Steps

When the inputs are equally spaced:

• outputs change by a constant amount → linear
• outputs change by a constant factor → exponential (the factor cannot be $0$ or $1$)

• Model A drops $30$ every $3$ units: linear, slope $-10$, so $A(x) = 200 - 10x$.
• Model B is multiplied by $0.8$ every $3$ units: exponential.

Different Domains

The sequence $g_n = 200(0.8)^n$ only has whole-number inputs. The function $f(x) = 200(0.8)^x$ also has values like $f(4.5)$, so it can describe what happens between the whole-number steps.

Pick the Model 🔽

Build the Function 🧮

1) Slope of the line through $(3, 14)$ and $(7, 30)$ 2) That line's value at $x = 0$ 3) The base $b$ of the exponential function through $(0, 50)$ and $(2, 72)$

🔢 Arithmetic & Geometric Sequences

Part 7 of 7 — Applications, Mastery & Exit Quiz

You now own both families end-to-end: classify, find any term from any known term, switch between recursive and explicit, compare their growth, and match them to linear and exponential functions. Let's apply it and then test it.

Quick Reference

⚠️ Three traps: use $n-1$ (not $n$) when counting from $a_1$; check whether the count starts at $0$ or $1$; and read percent language as a ratio ($5\%$ growth means $r = 1.05$).

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 $a_1 = 40000$, $d = 2500$:

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

Compare a $5\%$ raise each year instead, which is geometric with $r = 1.05$: year $10$ pays $40000(1.05)^9 \approx 62053$, still a little less, but year $11$ pays $40000(1.05)^{10} \approx 65156$ against the flat plan's $65,000. The percent raise grows by more each year, so it pulls ahead and stays ahead.

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 form you'd use.

Exit Quiz ✅

Answer all three to finish the lesson.