The sum of the first seven terms of the geometric sequence: 0.2, 0.8, 3.2, … is closest to:

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BlackTom AI · Accepted Answer

The sequence given is geometric with first term a1 = 0.2 and common ratio r = 0.8 / 0.2 = 4, since 0.8 ÷ 0.2 = 4 and 3.2 ÷ 0.8 = 4 confirms the ratio.
To find the sum of the first seven terms, use the geometric series sum formula: S_n = a1 * (r^n − 1) / (r − 1).
Here, S7 = 0.2 * (4^7 − 1) / (4 − 1) = 0.2 * (4^7 − 1) / 3.
Compute 4^7: 4^7 = 16384. Then 16384 − 1 = 16383.
Divide by 3: 16383 / 3 = 5461.
Multiply by 0.2: 0.2 * 5461 = 1092.2. Therefore the sum is exactly 1092.2.
Now evaluate the answer choices:
- Option a (2): This is far too small to be the sum of seven terms that escalate by a factor of 4 each time starting at 0.2. The sequence grows rapidly, so 2 is implausible.
- Option b (27.3): While larger than 2, this value is still an order of magnitude less than the actual sum; missing the exponential growth impact of r = 4 over seven terms.
- Option c (1092.2): This matches the computed sum exactly, aligning with the formula and the calculated 4^7 value.
- Option d (8.5): Like 2 and 27.3, this does not reflect the substantial accumulation from seven terms with such a large ratio; it undervalues the total.
In summary, only option c corresponds to the correct sum derived from the geometric-series formula with a1 = 0.2 and r = 4, yielding S7 = 1092.2.

The sum of the first seven terms of the geometric sequence: 0.2, 0.8, 3.2, … is closest to:

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