In a geometric sequence, the second term is 6 and the fifth term is 162. What is the sum of the first 6 terms?
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Correct answer
C. 728
Principle or equation
In a geometric sequence, the nth term is a_n = a_1 * r^(n-1). Given a_2 = a_1 r = 6 and a_5 = a_1 r^4 = 162, dividing gives r^3 = 27, so r = 3. Then a_1 = 2. The sum of the first n terms is S_n = a_1 (r^n - 1)/(r - 1) for r ≠ 1. Thus S_6 = 2(3^6 - 1)/(3 - 1) = 2(729 - 1)/2 = 728.
Why this answer is correct
From a_2 = 6 and a_5 = 162, we have a_1 r = 6 and a_1 r^4 = 162. Dividing the second by the first gives r^3 = 27, so r = 3. Then a_1 = 6/3 = 2. Sum of first 6 terms: S_6 = 2(3^6 - 1)/(3 - 1) = 2(729 - 1)/2 = 728.
Example
For a geometric sequence with a_1 = 1 and r = 2, the sum of the first 4 terms is 1(2^4 - 1)/(2 - 1) = 15.
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