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The sum, ∑ n = 1 7 n n + 1 2 n + 1 4 , is equal to

Correct answer

0

Step-by-step solution

Given, ∑ n = 1 7 n n + 1 2 n + 1 4 = 1 4 ∑ n = 1 7 2 n 3 + 3 n 2 + n Now using ∑ 1 + 2 + 3 + . . . + n = n n + 1 2 , ∑ 1 2 + 2 2 + 3 2 + . . . + n 2 = n n + 1 2 n + 1 6 and ∑ 1 3 + 2 3 + 3 3 + . . . + n 3 = n n + 1 2 2 , we can write = 1 4 2 7 · 8 2 2 + 3 7 · 8 · 15 6 + 7 · 8 2 1 4 2 × 49 × 16 + 28 × 15 + 28 1 4 1568 + 420 + 28 = = 504 .

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