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Question

By method of mathematical induction, the value of 13+33+53n terms=n2(xn21) is true for nN.
Then x is:

A
1
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B
2
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C
3
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D
4
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Solution

The correct option is B 2
Let P(n)=13+33+53n termsnN
so, it is also true for n=1
P(1)=1
From RHS, we have P(1)=1.(x1)=1
Hence on comparing we have x=2
Now, let P(n):13+33+53n termsnN is true
Check at n=n+1
P(n+1):13+33+53+(2n1)3+(2n+1)3=n2(2n21)+(2n+1)3
on expanding we get,
P(n+1)=2n4+8n3+11n2+6n+1 =(n+1)2[2(n+1)21]
So, P(n+1) is true.

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