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The sum to n terms of the series 

12 + 2.22 + 32 + 2.42 + 52 + ........ = n(n + 1)2/2 when n is even. When n is odd, the sum is

(a)  n2(n + 1)/2

(b)  n(n2 - 1)/2

(c)  n(n+1)2 (2n+1)

(d)  None of these

1 Answer

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Best answer

Correct option (a)  n2(n + 1)/2

Explanation:

12 + 2.22 + 32 + 2.42 + 52 + ........ + (2k - 1)2 + 2(2k)2

2k(2k + 1)2/2

Let n = 2k + 1 (odd)

12 + 2.22 + 32 + 2.42 + 52 + ........ + (2k - 1)2 + 2(2k)2 + (2k + 1)2

2k(2k + 1)2 /2 + (2k + 1)2

= (2k + 1)2 (2k + 2)/2 = n2(n + 1)/2

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