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Let Sₖ denote the sum of the first k terms of an arithmetic sequence whose first term is 1 and common difference is 1. Evaluate the sum: sum from k=2 to k=100 of (1/Sₖ).

1. 1/100
2. 101/100
3. 200/101
4. 99/101

Correct answer: 99/101

Solution

With Sₖ = k(k+1)/2, the sum becomes 2*sum(1/k - 1/(k+1)) from k=2 to 100, which telescopes to 2*(1/2 - 1/101) = 2*(99/202) = 99/101.

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