Geometric Series

Explanation

A geometric series is a sum of terms where each term is multiplied by the same number to get the next term.

• General form: a, ar, ar², ar³, …

• a = first term

• r = common ratio (multiplier)

Sum formula for finite geometric series (n terms):
Sₙ = a(1 − rⁿ) / (1 − r), r ≠ 1

Example:

• Series: 2 + 4 + 8 + 16

• a = 2, r = 2, n = 4

• S₄ = 2(1 − 2⁴) / (1 − 2) = 2(1 − 16)/(-1) = 2(-15)/(-1) = 30

Quiz

1. What is a geometric series?

2. Identify a and r in 3 + 6 + 12 + 24.

3. Sum first 5 terms of 1 + 2 + 4 + 8 + 16.

4. Can r be negative?

5. What happens if r = 1?

Answer Key

1. Series with a constant ratio between terms

2. a = 3, r = 2

3. S₅ = 1(1 − 2⁵)/(1 − 2) = 31

4. Yes

5. Series is just a repeated number: a + a + …