Geometric series · Year 13
In Year 13 you learn: sums to infinity.
A geometric series is the sum of a sequence that is multiplied by the same number, the common ratio r, each time. When r is between −1 and 1, the terms shrink so fast that the series has a sum to infinity.
Key facts
- Geometric: uₙ = arⁿ⁻¹ and Sₙ = a(1 − rⁿ)/(1 − r).
- If −1 < r < 1, a geometric series converges, with sum to infinity S∞ = a/(1 − r).
- a is the first term and r the common ratio: divide any term by the one before it.
Worked examples
Example 1
Find the sum of the first 8 terms of 3 + 6 + 12 + …
Answer: 765
Example 2
Find the common ratio of the geometric sequence 81, 27, 9, …
Answer: 1/3
Example 3
Find the 6th term of the geometric sequence 2, 6, 18, …
Answer: 486
Example 4
Evaluate Σ 3(2ʳ) from r = 1 to 10.
Answer: 6,138
Example 5
Find the sum to infinity of 50 + 25 + 12.5 + …
Answer: 100
Example 6
A geometric series has first term 24 and sum to infinity 36. Find the common ratio.
Answer: r = 1/3