In a geometric sequence, the fourth term is 24 and the seventh term is 192. What is the sum of the first 6 terms?
Show the answer and explanation
Correct answer
B. 186
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). Given a_m and a_n, r^(n-m) = a_n / a_m. The sum of the first k terms is S_k = a(1 - r^k)/(1 - r) for r ≠ 1.
Why this answer is correct
Let a be the first term and r the common ratio. We have a_4 = a r^3 = 24 and a_7 = a r^6 = 192. Dividing the second by the first gives r^3 = 8, so r = 2. Then a * 2^3 = 24 gives a = 3. The sum of the first 6 terms is S_6 = 3(1 - 2^6)/(1 - 2) = 3(1 - 64)/(-1) = 3 * 63 = 189.
Example
If a geometric sequence has a_2 = 6 and a_5 = 48, then r^3 = 8 so r = 2, a = 3, and S_4 = 3(1 - 16)/(1 - 2) = 45.
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