`1.2+2.2^2+3.2^3+.....+n.2^n=(n-1)2^(n-1)+2`

Explanation

For n=1
`L.H.S. =1.2=2`
`R.H.S. =(1-1).2^(1+1) +2`
`=0+2=2`
`:' " "L.H.S. =R.H.S.`
`:.` Given statement is true for n= 1
Let given statement be true for n=k
`:. 1. 2+2.2^(2)+3. 2^(3)+…..+ K.2^(K)`
`=(K-1).2^(k+1) +2 " ".......(1)`
for n =K+1
`1.2+2.2^(2)+3.2^(3)+......+k.2^(k)+(K+1).2^(k+1)`
`=(k-1).2^(k+1)+2+(k+1).2^(k+1)`
[from equation (1)]
`=[(k+1)+(K+1)].2^(k+1)+2`
`=2k.2^(k+1)+2`
`=k.2^(k+2)+2`
`rArr` Statement is also true for n=K+1.
Hence from the principle of mathematical induction the given statement is true for all natural numbers n.