Geometric Sequence & Series · 20 Problems

01 What Is a Geometric Sequence?

①

Definition

A geometric sequence is a list of numbers where each term is obtained by multiplying the previous term by a fixed number called the common ratio (r).

General Form

a, ar, ar², ar³, ar⁴, ...

To find the common ratio, divide any term by the one before it:

Common Ratio

r = a₂ ÷ a₁ = a₃ ÷ a₂ = aₙ₊₁ ÷ aₙ

Quick Example

In the sequence 5, 15, 45, 135, ..., what is r?

r = 15 ÷ 5 = 3 ✓

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📌

nth Term

aₙ = a₁ · r^(n−1)

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

∑

Finite Sum (r ≠ 1)

Sₙ = a₁(rⁿ − 1) / (r − 1)

Also written as a₁(1 − rⁿ) / (1 − r)

♾

Infinite Sum

S∞ = a₁ / (1 − r)

Only valid when |r| < 1 (converges)

√

Geometric Mean

b = √(a · c)

If a, b, c are in GP then b² = ac

02 Infinite Series & Convergence

②

When Does a Geometric Series Converge?

An infinite geometric series has a finite sum only when the terms get smaller and smaller, i.e. when the absolute value of r is less than 1.

Convergence Condition

|r| < 1 → Series CONVERGES → S∞ = a₁ / (1 − r)

Divergence Condition

|r| ≥ 1 → Series DIVERGES → No finite sum

Example — Infinite Sum

Find S∞ for 8 + 4 + 2 + 1 + ½ + …

r = ½, |r| < 1 ✓ → S∞ = 8 / (1 − ½) = 8 / (½) = 16

③

Key Properties to Remember

Positive r

All terms same sign

Negative r

Terms alternate signs

r = 1

Constant sequence

r = −1

Alternates ±a₁

0 < r < 1

Decreasing positive

r > 1

Increasing, diverges

Pattern Check

Sequence: 1, −2, 4, −8, 16, ... r = ?

r = −2 ÷ 1 = −2 (alternating, diverges)

0 / 20 answered

KEY Full Answer Key & Explanations

GeometricSequences & Series

Definition

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When Does a Geometric Series Converge?

Key Properties to Remember

Geometric
Sequences & Series