# If $a^{th}$ term of a series is $\left ( 2a+1 \right )2^{-a}$ then sum of its infinite term is Option 1) 10 Option 2) 8 Option 3) 5 Option 4) 0

As we learnt in

Sum of infinite terms of an AGP -

$S_{\infty }= \frac{a}{1-r}+\frac{dr}{\left ( 1-r \right )^{2}}\\here \: \left | r \right |< 1$

- wherein

$a\rightarrow$ first term

$d\rightarrow$ common diff. of a AP

$r\rightarrow$ common ratio of a GP

$S_{\infty }=\frac{3}{2}+\frac{5}{2^{2}}+\frac{7}{2^{3}}+...........\infty \\*\\*\frac{S_{\infty }}{2}=\frac{3}{2^{2}}+\frac{5}{2^{3}}+...........\infty \\*\\*\frac{S_{\infty }}{2}=\frac{3}{2}+\frac{1}{2}+\frac{1}{2^{2}}+...........\infty \\*\\*\frac{S_{\infty }}{2}=\frac{3}{2}+1\\*\\*S_{\infty }=5$

Option 1)

10

Incorrect

Option 2)

8

Incorrect

Option 3)

5

Correct

Option 4)

0

Incorrect

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