Question

Find the sum of n terms of the series $1.3+{3.3}^2+{5.3}^3+{7.3}^4+{9.3}^5+.....$

Answer 1

$\Rightarrow u_n=(2n-1)3^n$

$==(An+B)3^n-\left\{A\right(n-1)+B\}{3}^{n-1}$

Divide by $3^{n-1}$ and equate coefficients, thus we obtain $A=3,B=-3,$

$\therefore u_n=(3n-3)\cdot 3^n-\{3(n-1)-3\}{3}^{n-1}$

$u_{n-1}=\left\{3\right(n-1)-3\}{3}^{n-1}-\left\{3\right(n-2)-3\}{3}^{n-2}$

.

.

.

$u_1=0+3$

$S_n=(3n-3)3^n+3$

$=(n-1)3^{n+1}+3$