In a geometric sequence, the second term is 12 and the fifth term is 324. What is the sum of the first 5 terms?
Show the answer and explanation
Correct answer
A. 484
Principle or equation
For a geometric sequence with first term a and common ratio r, the nth term is a_n = a r^(n-1). The sum of the first n terms is S_n = a(1 - r^n)/(1 - r) for r ≠ 1.
Why this answer is correct
Given a_2 = ar = 12 and a_5 = ar^4 = 324. Dividing the second by the first gives r^3 = 27, so r = 3. Then a = 12/3 = 4. The sum of the first 5 terms is S_5 = 4(1 - 3^5)/(1 - 3) = 4(1 - 243)/(-2) = 4(-242)/(-2) = 4(121) = 484.
Example
If a geometric sequence has a_2 = 6 and a_5 = 162, then r^3 = 27, r = 3, a = 2, and S_5 = 2(1 - 3^5)/(1 - 3) = 2(1 - 243)/(-2) = 242.
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