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

1.	`1.3+2.3^2+3.3^3+..............+n.3^n=((2n-1)3^(n+1)+3)/4`
Answer» 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)+â€¦â€¦.+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

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