The sum of the first seven terms of the geometric sequence: 0.2, 0.8, 3.2, … is closest to:单项选择题
A
a. 8.5
B
b. 2
C
c. 1092.2
D
d. 27.3

Question

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

BlackTom AI · Accepted Answer

This is a geometric sequence with first term a1 = 0.2 and common ratio r = 0.8 / 0.2 = 4. The relevant knowledge: the sum of the first n terms of a geometric sequence is S_n = a1 (r^n − 1) / (r − 1) for r ≠ 1.

Step-by-step reasoning:
1. Identify parameters: a1 = 0.2, r = 4, n = 7.
2. Compute r^7: 4^2 = 16, 4^3 = 64, 4^4 = 256, 4^5 = 1024, 4^6 = 4096, 4^7 = 16384.
3. Plug into the sum formula:
   - r^7 − 1 = 16384 − 1 = 16383.
   - a1 (r^7 − 1) = 0.2 × 16383 = 16383 / 5 = 3276.6.
   - r − 1 = 4 − 1 = 3.
   - S7 = 3276.6 / 3 = 1092.2.

Verification by listing terms and adding:
- Terms: t1 = 0.2, t2 = 0.8, t3 = 3.2, t4 = 12.8, t5 = 51.2, t6 = 204.8, t7 = 819.2.
- Cumulative sum: 0.2 + 0.8 = 1.0; +3.2 = 4.2; +12.8 = 17.0; +51.2 = 68.2; +204.8 = 273.0; +819.2 = 1092.2.

Why the correct answer is correct:
The formula directly gives S7 = 1092.2 and the term-by-term sum confirms this numeric result. The sequence grows rapidly because r = 4, so later terms dominate the sum; this explains the large total.

Why the other choices are incorrect:
- a. 8.5 and b. 2 are far too small (they would correspond to only the first few terms, not seven).
- d. 27.3 is also much smaller than the true sum (even after five terms the sum is 68.2), so 27.3 cannot be correct.

Conclusion: The sum of the first seven terms is 1092.2, matching answer c.

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

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