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

Question

BlackTom AI · Accepted Answer

For a geometric sequence with first term a and common ratio r (r ≠ 1), the sum of the first n terms is S_n = a (r^n − 1) / (r − 1).

1) Relevant knowledge points and concepts
- Definition of a geometric sequence: each term is obtained by multiplying the previous term by a constant ratio r.
- Formula for the sum of the first n terms: S_n = a (r^n − 1) / (r − 1) when r ≠ 1.
- Ability to compute powers and perform basic arithmetic.

2) Complete reasoning process step by step
- Identify the first term a and the common ratio r from the sequence 0.2, 0.8, 3.2, ...:
  a = 0.2. r = 0.8 / 0.2 = 4 (also 3.2 / 0.8 = 4).
- We need the sum of the first seven terms, so n = 7. Use S_7 = a (r^7 − 1) / (r − 1).

3) Calculations with intermediate steps
- Compute r^7: 4^7 = 4^5 · 4^2 = 1024 · 16 = 16384.
- r^7 − 1 = 16384 − 1 = 16383.
- Multiply by a: a (r^7 − 1) = 0.2 × 16383 = 3276.6.
- Denominator r − 1 = 4 − 1 = 3.
- Divide: S_7 = 3276.6 / 3 = 1092.2.

As a check, list the first seven terms and their running sum:
t1 = 0.2
t2 = 0.8 → sum 1.0
t3 = 3.2 → sum 4.2
t4 = 12.8 → sum 17.0
t5 = 51.2 → sum 68.2
t6 = 204.8 → sum 273.0
t7 = 819.2 → final sum 1092.2

4) Why the correct answer is correct
The closed-form sum formula was applied with the correct a = 0.2 and r = 4, and arithmetic yields S_7 = 1092.2, exactly matching the computed running-sum check.

5) Why the other options are incorrect
- b. 27.3 and d. 8.5 are far too small compared to the rapidly growing terms (by term 7 we already have 819.2).
- c. 2 is impossible since the sum after three terms is already 4.2, so 2 is too small.

6) Conclusion
The sum of the first seven terms is 1092.2, so the correct choice is a. 1092.2, which matches the provided answer.

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

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