1.2 +2.2² + 3.2² +…. +n.2ⁿ = (n – 1) 2ⁿ⁺¹ + 2.

P(1_ is trueAssuming P(k) is true.
1.2 +2.2² + 3.2² +…. +k.2^k = (k – 1) 2 ^k+1 + 2
To prove P(k+1) is true using P(k) lies
1.2 +2.2² + 3.2² +…. +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 + 2.2² + 3.2² +………..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.