In a geometric sequence, the second term is 6 and the fifth term is 162. What is the sum of the first 5 terms?
Show the answer and explanation
Correct answer
A. 242
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) when r ≠ 1.
Why this answer is correct
Given a_2 = a r = 6 and a_5 = a r^4 = 162. Divide: (a r^4)/(a r) = r^3 = 162/6 = 27, so r = 3. Then a * 3 = 6 → a = 2. Sum first 5: S_5 = 2(1 - 3^5)/(1 - 3) = 2(1 - 243)/(-2) = 242.
Example
If a_1 = 1 and r = 2, then S_4 = 1(1 - 2^4)/(1 - 2) = 15.
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