Precalculus — Chapter 12

Geometric Sequences & Series

Q: What is the formula for the sum of the first n terms of a geometric series?

The partial sum of a geometric series is Sₙ = a₁(1 − rⁿ)/(1 − r) when r ≠ 1. If r = 1, then Sₙ = n·a₁ (just multiply the first term by n). Example: sum of first 6 terms of 3 + 6 + 12 + ... (r=2): S₆ = 3(1−2⁶)/(1−2) = 3(1−64)/(−1) = 3(63) = 189.

Common ratio, nth term formula, partial sums, and infinite series convergence — with real-world applications.

Key Formulas

nth Term

aₙ = a₁ · r^(n−1)

a₁ = first term, r = common ratio, n = term number

Common Ratio

r = aₙ₊₁ / aₙ

Any consecutive term divided by the one before it

Partial Sum (first n terms)

Sₙ = a₁(1 − rⁿ) / (1 − r)

Valid when r ≠ 1. If r = 1: Sₙ = n·a₁

Infinite Sum (|r| < 1)

S∞ = a₁ / (1 − r)

Only exists when −1 < r < 1 (series converges)

Geometric vs. Arithmetic — Quick Check

	Arithmetic	Geometric
Pattern	Add constant d each term	Multiply by constant r each term
Test	Differences are constant	Ratios are constant
Example	3, 7, 11, 15 ... (d=4)	3, 6, 12, 24 ... (r=2)
nth term	a₁ + (n−1)d	a₁ · r^(n−1)
Graph shape	Linear	Exponential

Worked Examples

Example 1: Find the 10th term

Sequence: 2, 6, 18, 54, ...

r = 6/2 = 3. a₁ = 2, n = 10

a₁₀ = 2 · 3^(10−1) = 2 · 3⁹ = 2 · 19683

a₁₀ = 39,366

Example 2: Sum of first 8 terms

Sequence: 5 + 10 + 20 + 40 + ... (r = 2, a₁ = 5)

S₈ = 5(1 − 2⁸) / (1 − 2)

= 5(1 − 256) / (−1)

= 5(−255) / (−1)

= 1275

Example 3: Infinite series

Find the sum: 16 + 8 + 4 + 2 + 1 + ...

r = 8/16 = 1/2. |r| = 0.5 < 1 → converges

S∞ = 16 / (1 − 1/2) = 16 / (1/2)

= 32

Example 4: Real-world — Bacteria growth

Bacteria doubles every hour. Start: 500. How many after 8 hours?

a₁ = 500, r = 2, n = 9 (after 8 doublings, we're on term 9)

a₉ = 500 · 2^8 = 500 · 256

= 128,000 bacteria

Example 5: Repeating decimal as infinite series

Write 0.333... as a fraction

= 3/10 + 3/100 + 3/1000 + ...

a₁ = 3/10, r = 1/10

S∞ = (3/10) / (1 − 1/10) = (3/10) / (9/10)

= 3/10 × 10/9 = 1/3 ✓

Geometric Means

The geometric mean of two numbers a and b is √(ab) — the number that could be inserted between them to form a geometric sequence.

Insert 2 geometric means between 4 and 108

Sequence: 4, ?, ?, 108 → a₁ = 4, a₄ = 108, n = 4

a₄ = a₁ · r³ → 108 = 4r³ → r³ = 27 → r = 3

Sequence: 4, 12, 36, 108

Convergence of Infinite Geometric Series

|r| < 1 (e.g., r = 0.5)

CONVERGES

Sum = a₁/(1−r). The terms approach 0 fast enough to sum to a finite number.

|r| = 1 (e.g., r = 1 or r = −1)

DIVERGES

r=1: terms never decrease. r=−1: terms alternate and never settle.

|r| > 1 (e.g., r = 2)

DIVERGES

Terms grow without bound — the sum is infinite.

Frequently Asked Questions

What is the formula for the nth term of a geometric sequence?

The nth term of a geometric sequence is aₙ = a₁ · r^(n−1), where a₁ is the first term and r is the common ratio. To find r, divide any term by the preceding term: r = aₙ₊₁/aₙ.

When does an infinite geometric series converge?

An infinite geometric series converges (has a finite sum) when |r| < 1 (the common ratio is between −1 and 1). The sum is S∞ = a₁/(1 − r). If |r| ≥ 1, the series diverges (no finite sum). Example: 1 + 1/2 + 1/4 + ... has r = 1/2, so S∞ = 1/(1 − 1/2) = 2.

What is the formula for the sum of the first n terms of a geometric series?