Geometric Series

A geometric series is a series whose related sequence is geometric. It results from adding the terms of a geometric sequence . Example 1: Finite geometric sequence: 1 2 , 1 4 , 1 8 , 1 16 , ... , 1 32768 Related finite geometric series: 1 2 + 1 4 + 1 8 + 1 16 + ... + 1 32768 Written in sigma notation: ∑ k = 1 15 1 2 k Example 2: Infinite geometric sequence: 2 , 6 , 18 , 54 , ... Related infinite geometric series: 2 + 6 + 18 + 54 + ... Written in sigma notation: ∑ n = 1 ∞ ( 2 ⋅ 3 n − 1 ) Finite Geometric Series To find the sum of a finite geometric series, use the formula, S n = a 1 ( 1 − r n ) 1 − r , r ≠ 1 , where n is the number of terms, a 1 is the first term and r is the common ratio . Example 3: Find the sum of the first 8 terms of the geometric series if a 1 = 1 and r = 2 . S 8 = 1 ( 1 − 2 8 ) 1 − 2 = 255 Example 4: Find S 10 , the tenth partial sum of the infinite geometric series 24 + 12 + 6 + ... . First, find r . r = a 2 a 1 = 12 24 = 1 2 Now, find the sum: S 10 = 24 ( 1 − (...