Prove the following by using the principle of mathematical induction for all `n in N`:`1. 2+2. 2^2+3. 2^2+...+n .2^n=(n-1)2^(n+1)+2`

Prove the following by using the principle of mathematical induction f

1.	Prove the following by using the principle of mathematical induction for all `n in N`:`1. 2+2. 2^2+3. 2^2+...+n .2^n=(n-1)2^(n+1)+2`
Answer» `P(k): 12+22^(2)+32^(3)+…+k2^(k)=(k-1)2^(k+1)+2`. …(i) Now, `{12+22^(2)+32^(3)+…+k2^(k)}+(k+1)2^(k+1)` ` ={(k-1)2^(k+1)+2}+(k+1)2^(k+1)= 2k2^(k+1)+2 = k2^(k+1)+2`.