Sum of a geometric sequence with first term a, common ratio r, and number of terms n:
When r ≠ 1: S_n = a(1 − rⁿ)/(1 − r) = a(rⁿ − 1)/(r − 1)
When r = 1: S_n = na
Example ① Find the sum of 2 + 6 + 18 + … (10 terms).
Solution: First term 2, common ratio 3, n=10.
S = 2(3¹⁰ − 1)/(3 − 1) = 3¹⁰ − 1 = 59048
Solution: First term 2, common ratio 3, n=10.
S = 2(3¹⁰ − 1)/(3 − 1) = 3¹⁰ − 1 = 59048
Key Point
- Do not forget to consider the case r=1 separately
- When r>1, (rⁿ−1)/(r−1) is easier to calculate
Practice Problems
- Find the sum of 1 + 1/2 + 1/4 + … (8 terms).
Answer and Explanation
- Answer:255/128
The common ratio is 1/2. S = 1·(1−(1/2)⁸)/(1−1/2) = 2(1−1/256) = 255/128.