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Let S1, S2, S3,..., Sn denote the sums of infinite geometric progressions whose first terms are 1, 2, 3,..., n and whose common ratios are 1/2, 1/3, 1/4,..., 1/(n+1), respectively. Then the value of S1² + S2² + S3² +... + Sn² is

1. 1/3 [n(2n+1)(4n+1)-3]
2. 1/3 [n(2n+1)(4n+1)+3]
3. 1/3 [n(2n-1)(4n+1)-3]
4. None of these

Correct answer: None of these

Solution

Each S_k = k/(1 - 1/(k+1)) = k+1, so the required sum is sum_{k=1}^{n} (k+1)^2 = (1/6)n(2n^2+9n+13). Checking the listed forms (e.g. at n=2 the true value is 13) none of them agree, so the answer is None of these.

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