Question

Find the sum of n terms of the series $\frac{1.2}{\lfloor 3}+\frac{{2.2}^2}{\lfloor 4}+\frac{{3.2}^3}{\lfloor 5}+\frac{{4.2}^4}{\lfloor 6}+....$

Answer 1

$t_n=\frac{n\cdot 2^n}{(n+2)!}$

$=\frac{(n+1)2^n}{(n+2)!}$

$=(\frac{1}{(n+1)!}-\frac{1}{(n+2)!})2^n$

$=\frac{2^n}{(n+1)!}-\frac{2^{n+1}}{(n+2)!}$

$t_n=\frac{2^n}{(n+1)!}-\frac{2^{n+1}}{(n+2)!}$

$t_{n-1}=\frac{2^{n-1}}{n!}-\frac{2^n}{(n+1)!}$

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.$t_1=\frac{2}{2!}-\frac{2^2}{3!}$

$S_n=1-\frac{2^{n+1}}{(n+2)!}$