If S = ∑ n = 1 9999 1 n + n + 1 n 4 + n + 1 4 , then the value of S is equal to

Question

If S = ∑ n = 1 9999 1 n + n + 1 n 4 + n + 1 4 , then the value of S is equal to, 9, 99, 999, 9999

MARKS App · Accepted Answer

Given, S = ∑ n = 1 9999 1 n + n + 1 n 4 + n + 1 4 = ∑ n = 1 9999 1 n + n + 1 n 4 + n + 1 4 × n + 1 4 - n 4 n + 1 4 - n 4 = ∑ n = 1 9999 n + 1 4 - n 4 n + n + 1 n + 1 - n = ∑ n = 1 9999 n + 1 4 - n 4 = 2 1 4 - 1 + 3 1 4 - 2 1 4 + 4 1 4 - 3 1 4 + … + 10000 1 4 - 9999 1 4 = 10000 1 4 - 1 = 10 - 1 = 9

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