What You Need to Know
This chapter covers two interconnected ideas: geometric sequences (ordered lists of numbers with a pattern) and geometric series (summing those numbers). You'll learn formulas to find specific terms, sum first n terms, and determine when an infinite series converges to a finite number.
Geometric Sequences
Each term is the previous term multiplied by a constant ratio. Use the nth term formula to find any term directly.
Geometric Series
The sum of a geometric sequence. Two formulas depending on whether r > 1 or r < 1; sum to infinity when |r| < 1.
Convergence
An infinite geometric series converges to a finite limit only when |r| < 1. This is a critical exam concept.
Geometric Sequences
What is a Geometric Sequence?
A geometric sequence is a list of numbers where each term is found by multiplying the previous term by a fixed number. That fixed number is called the common ratio, usually written as r.
Example 1: The sequence 2, 6, 18, 54, 162, …
2 → ×3 → 6 → ×3 → 18 → ×3 → 54 → …
Here, the first term is 2 (often called a) and the common ratio is 3 (so r = 3).
Example 2: The sequence 2, −1, 0.5, −0.25, 0.125, …
2 → ×(−0.5) → −1 → ×(−0.5) → 0.5 → ×(−0.5) → −0.25 → …
First term a = 2, common ratio r = −0.5. Notice r can be negative!
The key insight: once you know the first term and the common ratio, you can write down any term in the sequence instantly — no need to multiply your way through all the previous terms.
The nth Term Formula
For a geometric sequence with first term a and common ratio r, the nth term is:
Why Does This Formula Work?
Imagine the first term is a. To get the second term, multiply by r once: ar. To get the third term, multiply by r twice: ar². To get the nth term, multiply by r exactly (n − 1) times: arn−1. That's where the exponent comes from.
Quick verification: In our first example (2, 6, 18, 54, …) with a = 2 and r = 3:
- u₁ = 2 × 3⁰ = 2 × 1 = 2 ✓
- u₂ = 2 × 3¹ = 2 × 3 = 6 ✓
- u₃ = 2 × 3² = 2 × 9 = 18 ✓
- u₄ = 2 × 3³ = 2 × 27 = 54 ✓
- u₁₀ = 2 × 3⁹ = 39,366 ✓
Finding a and r from Two Known Terms
Often, exams give you two terms in a sequence and ask you to find the first term and common ratio. The method is: write two equations using the formula, then solve them simultaneously.
Worked Example: Finding a and r
The 6th term is 486, and the 40th term is 39,366. Find a and r.
This is a classic exam pattern. We're given two terms, not the first term and ratio directly.
Solution:
Set up two equations:
u₆ = ar⁵ = 486 ... (1)
u₄₀ = ar³⁹ = 39,366 ... (2)
Divide equation (2) by equation (1):
ar³⁹ ÷ ar⁵ = 39,366 ÷ 486
r³⁴ = 81
r³⁴ = 3⁴
r = 3
Substitute r = 3 back into equation (1):
a × 3⁵ = 486
a × 243 = 486
a = 2
Key insight: