Section 11.3

Geometric Sequences

Geometric sequences model anything that grows or shrinks by a percentage — compound interest, population growth, depreciation. On Exam 4, you'll see both finite and infinite series problems, and the infinite formula is also how you convert repeating decimals to fractions.

⚡ Quick Summary

A geometric sequence is a pattern where you multiply by the same number each time instead of adding. Imagine a chain letter: you send it to 3 people, each of them sends it to 3 more, and so on — the number of letters grows as 1, 3, 9, 27, ... That multiplier (3 in this case) is called the common ratio $r$ . If $r$ is a fraction (like $\frac{1}{2}$ ), the terms shrink instead of grow — like a bouncing ball that goes half as high each time.

Explicit (general) term

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

jumps directly to the $n$ th term; $a_1$ is the first term, $r$ is the common ratio

Recursive definition

a_n = r \cdot a_{n-1}, \quad a_1 \text{ given}

each term equals the previous term times $r$

📋 Before You Start

Tap any item if you need a refresher:

Plain-English version

The formula $a_n = a_1 \cdot r^{n-1}$ tells you any term directly. The exponent is $n-1$ (not $n$ ) because the first term hasn't been multiplied yet — it's your starting point. So the 5th term means you've multiplied by $r$ a total of 4 times from the start.

Here's where it gets cool: if the ratio $r$ is a fraction between $-1$ and $1$ (meaning $|r| < 1$ ), the terms keep getting smaller and smaller, approaching zero. When you add up infinitely many of these shrinking terms, the total actually settles on a finite number! That infinite sum is $S = \frac{a_1}{1 - r}$ . This is how we can convert repeating decimals like $0.333...$ into exact fractions.

If $|r| \geq 1$ , the terms don't shrink — they stay the same size or get bigger — so the infinite sum just keeps growing forever. We say it diverges. On an exam, always check $|r|$ first before using the infinite sum formula. If $|r| \geq 1$ , just write "diverges" and you're done.

Key Formulas

Explicit (general) term

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

jumps directly to the $n$ th term; $a_1$ is the first term, $r$ is the common ratio

Recursive definition

a_n = r \cdot a_{n-1}, \quad a_1 \text{ given}

each term equals the previous term times $r$

Finite geometric series

S_n = a_1 \cdot \frac{1 - r^n}{1 - r}, \quad r \neq 1

sum of the first $n$ terms

Infinite geometric series

S = \frac{a_1}{1 - r}, \quad |r| < 1

only converges when $|r| < 1$ ; otherwise the series diverges

Key Takeaways

✓Explicit formula: $a_n = a_1 \cdot r^{n-1}$ — the exponent is $n-1$ , not $n$
✓Infinite series converges only when $|r| < 1$ ; use $S = \frac{a_1}{1 - r}$
✓A negative $r$ makes terms alternate in sign — check $|r|$ (absolute value) for convergence
✓For finite sums, use $S_n = a_1 \cdot \frac{1 - r^n}{1 - r}$ — the two negatives often cancel

⚠️ Common Mistakes

✗Using $r^n$ in the explicit formula instead of $r^{n-1}$ — the first term is $a_1 \cdot r^0 = a_1$ , not $a_1 \cdot r^1$
✗Forgetting to check $|r| < 1$ before using the infinite series formula — if $|r| \geq 1$ , the series diverges and the formula doesn't apply
✗Ignoring the sign of $r$ when checking convergence — $r = -\frac{1}{2}$ still converges because $|{-\frac{1}{2}}| = \frac{1}{2} < 1$
✗In bouncing ball problems, forgetting to double the bounce heights — after the initial drop, each bounce is traveled twice (up then down)
✗Dividing by $1 - r$ incorrectly when $r$ is a fraction — $\frac{a_1}{1 - \frac{1}{3}} = \frac{a_1}{\frac{2}{3}}$ means multiply $a_1$ by $\frac{3}{2}$ , not divide

Next: Section 9.3 Builds on: 11.1

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11.2 Arithmetic Sequences

9.3 Summation Notation