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Question

Sum of n terms of the following series 13+33+53+73+............ is


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Solution

Finding the sum of n terms of the following series

The given series is: 13+33+53+73+............

Here, the nth term of the series can be given as

Tn=2n-13......(1)

We know that

a-b3=a3-3a2b+3ab2-b3

Then, Equation (1) becomes

Tp=2p-13

Tp=2p3-3×2p2×1+3×2p×12-13=8p3-12p2+6p-1

Now, the sum Sn = Sum of 'n' terms =∑p=1nTp

Since,

Sn=∑p=1n8p3-12p2+6p-1=8×∑p=1np3-12×∑p=1np2+6×∑p=1np-∑p=1n1=8×n2n+124-12×nn+12n+16+6×nn+12-n=2n2n+12-2n2+n2n+1+3n2+n-n=2n2n2+2n+1-22n3+3n2+n+3n2+3n-n=2n4+4n3+2n2-4n3-6n2-2n+3n2+3n-n=2n4-n2=n22n2-1

Hence, the correct answer is n22n2-1.


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