`1.3+2.3^2+3.3^3+..............+n.3^n=((2n-1)3^(n+1)+3)/4`

Explanation

For n=1
`L.H.S.=1.3=3` ,
`R.H.S. =((2-1).3^(2)+3)/(4)`
`=(9+3)/(4)=3`
` :. L.H.S. =R.H.S.`
Therefore given statements is true for n=1
Let the statement be true for n=k.
`:. 1.3+2.3^(2)+3.3^(3)+…….+k.3^(k)`
`=((2k -1)3^(k+1)+3)/(4)`
For n =k +1
`1.3+2.3^(2)+3.3^(3)+.....+k.3^(k)+(k+1).3^(k+1)`
`=((2k-1)3^(k+1)+3)/(4) +(k+1).3^(k+1)`
[From equation (a)]
`=((2k-1)3^(k+1)+3+4(k+1).3^(K=1))/(4)`
`=((2k-1+4K+4).3^(k+1)+3)/(4) `
`=((6k+3).3^(k+1)+3)/(4)`
`=(3(2k+1).3^(k+1)+3)/(4)`
`=((2K+1).3^(k+2)+3)/(4)`
`rArr` statement is also true for n=k+1
Hence form the principle of mathematical induction the given statement is true for all natural numbers n