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Question: The value of \(\frac{1.2}{3!}\)+ \(\frac{2.2^{2}}{4!}\)+ \(\frac{3.2^{3}}{5!}\)…..+ \(\frac{15.2^{15...

The value of 1.23!\frac{1.2}{3!}+ 2.224!\frac{2.2^{2}}{4!}+ 3.235!\frac{3.2^{3}}{5!}…..+ 15.21517!\frac{15.2^{15}}{17!}=

A

1– 21517!\frac{2^{15}}{17!}

B

1– 21617!\frac{2^{16}}{17!}

C

1−16.21519!1 - 16.\frac{2^{15}}{19!}

D

None of these

Answer

1– 21617!\frac{2^{16}}{17!}

Explanation

Solution

Tr = r.2r(r+2)!\frac{r.2^{r}}{(r + 2)!}= 2r(r+1)!\frac{2^{r}}{(r + 1)!}– 2r+1(r+2)!\frac{2^{r + 1}}{(r + 2)!}

\ ∑r=115Tr\sum_{r = 1}^{15}T_{r}=22!\frac{2}{2!}– 21617!\frac{2^{16}}{17!}= 1– 21617!\frac{2^{16}}{17!}