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Using the principle of mathematical indu...

Using the principle of mathematical induction prove that `:` `1. 3+2. 3^2+3. 3^3++n .3^n=((2n-1)3^(n+1)+3)/4^` for all `n in N` .

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Assume P(k) is true, i.e, `(1.3)+(2.3)^2+(3.3)^3+.......+(k.3)^k=4(2k−1)3^(k+1)+3​ ...(1)`
Now let us prove that P(k+1) is true,
LHS: `1.3+(2.3)^2+(3.3)^3+...(k.3)^k+[(k+1).3]^(k+1)` substitute eq (1),
`=(((2k−1)3^(k+1)+3)​/4+[(k+1).3]^(k+1)`
...
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