. The cubes of the natural number are grouped as


The cubes of the natural number are grouped as 13, (23, 33) , (43, 53, 63),………. Let Sn denote the sum of the cubes in the nth group, then 8Sis divisible by

A. n2

B. n3

C. n2+1

D. n2+3

Best Answer

The numbers in the rth group lie in the rth row of the triangle

the rth row of the triangle

Note that the rth row contains r elements. Therefore, the number of elements in the first r

Rows = 1 + 2 + 3 + …….. + r = 1/2 r(r+1) = k (say).

Thus, sum of all the elements in the first r rows of the triangle

Tr = 13 + 22 + 33 + …… + k3 = 1/2 k2 (k + 1)2

= 1/4 . 1/4 r2 (r + 1)2 [1/2(r2 + r + 2]2

= 1/64 r2 (r + 1)2 (r2 + r + 2)2

Sn = Tn – Tn – 1

= 1/64 n3 (n + 1)2 (n2 + n + 2)2 – 1/64 (n – 1)2n2((n – 1)2 + (n – 1)+2)2

8S­n =n3 (n2 + 1) (n2 +3)

8Sn is divisible by n2, n3, n2 + 1 and n2 + 3.

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