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Question

If 1+1+22+1+2+33+.... to n terms is S, then S is equal to

(a) n (n+3)4

(b) n (n+2)4

(c) n (n+1) (n+2)6

(d) n2

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Solution

a n (n+3)4

Let Tn be the nth term of the given series.

Thus, we have:

Tn=1+2+3+4+5+...+nn=nn+12n=n2+12

Now, let Sn be the sum of n terms of the given series.

Thus, we have:

Sn= k=1nk2+12 Sn= k=1nk2+n2 Sn= nn+14+n2 Sn= n2n+12+1 Sn= n2n+32 Sn= nn+34

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