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Let Sₖ be the sum of the first k terms of the arithmetic sequence with first term 1 and common difference 1. Then the value of the sum from k = 2 to k = 100 of (1 / Sₖ) is equal to:
- 99/100
- 101/100
- 1/100
- 100/101
Correct answer: 99/100
Solution
Sₖ = k(k+1)/2, so 1/Sₖ = 2/(k(k+1)) = 2(1/k - 1/(k+1)). Summing from k=2 to 100 gives a telescoping series: 2*(1/2 - 1/101) = 2*(101-2)/(2*101) = 99/101. Wait — let me recheck: 2*(1/2 - 1/101) = 1 - 2/101 = 99/101. The correct answer is 99/101.
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