Prove the following by using the principle of mathematical induction for all 1.2+2.2^2+3.2^2+...+n.2^n

We can write the given statement as

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

If n = 1 we get

P (1): 1.2 = 2 = (1 – 1) 2¹⁺¹ + 2 = 0 + 2 = 2

Which is true.

Consider P (k) be true for some positive integer k

1.2 + 2.2² + 3.2² + … + k.2^k = (k – 1) 2^{k + 1} + 2 … (i)

Now let us prove that P (k + 1) is true.

Here

1.2 + 2.2² + 3.2² + … + k.2^k + (k + 1) 2^{k + 1} 

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

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

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

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

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

P (k + 1) is true whenever P (k) is true.

Therefore, by the principle of mathematical induction, statement P (n) is true for all natural numbers i.e. n.