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Question

Let Sn denote the sum of the cubes of the first n natural numbers and Sn denote the sum of the first natural numbers. Then r=1n SrSr equals
(a) nn+1n+26

(b) nn+12

(c) n2+3n+22

(d) None of these

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Solution

Let Sn denote the sum of the cubes of the first n natural numbers and Sn denotes the sum of the first n natural number 1.
Then r=1n SrSr=?
Since Sn= nn+122 ; formula for sum of cubes.
Sn=nn+12 ; formula form sum of first n natural numbers.
r=1n SrSr=r=1n rr+122rr+12= 12 r=1n r r+1 = 12 r=1n r2 + r=1n r = 12 nn+1 2n + 16 + nn+12 = 12 nn+12 2n+13+1 = 14 n n+1 2n+1+33 = 14 n n+1 2n+43 = nn+1 2n+212
Hence, r=1n SrSn= nn+1 n+26
Hence, the correct answer is option A.

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