Answer :

P(1_ is trueAssuming P(k) is true.

1.2 +2.2

^{2}+ 3.2

^{2}+…. +k.2

^{k}= (k – 1) 2

^{k+1}+ 2

To prove P(k+1) is true using P(k) lies

1.2 +2.2

^{2}+ 3.2

^{2}+…. +k.2

^{k}= (k + 1) 2

^{k+1}= (k)2

^{k+1+1}+ 2

= k.2

^{k+2}+ 2 ……………(1)

L.H.S of (1)

1.2 + 2.2

^{2}+ 3.2

^{2}+………..k.2

^{k}+ (k+1)2

^{(k+1)}(k-1)2

^{k+1}+ 2 + (k +1) 2

^{k+1 }= 2

^{k+1}(k+1+k-1)+2

= 2

^{k+1}2k + 2

= 2

^{k+2}.k + 2

= k.2

^{k+2}+ 2

Which is the R.H.s of (1)

P(K+1) is true.

By the Principle of mathematical induction, P(n) is true for all values of n where n N

Hence proved.

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