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Q.
The sum of the series $\frac{12}{2 !}+\frac{28}{3 !}+\frac{50}{4 !}+\frac{78}{5 !}+\ldots \ldots \ldots \ldots$ is
Binomial Theorem
Solution:
$t _{ n }=12+\frac{ n -1}{2}[2 \times 16+( n -2) \times 6]=3 n ^{2}+7 n +2$
So, the general term of the given series
$T _{ n }=\frac{3 n ^{2}+7 n +2}{( n +1) !}=\frac{3\left( n ^{2}+2 n +1\right)+ n -1}{( n +1) !}$
$=\frac{3(n+1)}{n !}+\frac{n+1-2}{(n+1) !}=\frac{3}{(n-1) !}+\frac{3}{n !}+\frac{1}{n !}-\frac{2}{(n+1) !}$
$=\frac{3}{(n-1) !}+\frac{4}{n !}-\frac{2}{(n+1) !}$
$\therefore $ Sum $= \displaystyle\sum_{n=1}^{\infty} T_{n}=3 e+4(e-1)-2(e-2)=5 e$
