# In an infinite Geomatric progression ,each term is equal to twice the sum of all the terms that follow it. if the sum of first two terms is 12, the sum of entire progression is Option 1) $\frac{9}{2}$ Option 2) $\frac{27}{2}$ Option 3) $\frac{88}{7}$ Option 4) $15$

As we learnt in

Sum of infinite terms of a GP -

$\\a+ar+ar^{2}+- - - - -= \frac{a}{1-r}\\here \left | r \right |<1$

- wherein

$a\rightarrow$ first term

$r\rightarrow$ common ratio

Given

$\\a=2\left ( ar+ar^{2+} +ar^{3}+........\infty \right )\\*\\* Thus, 1=\frac{2r}{1-r}\\*\\*1-r=2r\\*\\r=\frac{1}{3}$

Also a+ar=12

4a/3=12

a=9

So, $S_\infty =\frac{a}{1-r}=\frac{9}{1-\frac{1}{3}}=\frac{27}{2}$

Option 1)

$\frac{9}{2}$

Incorrect

Option 2)

$\frac{27}{2}$

Correct

Option 3)

$\frac{88}{7}$

Incorrect

Option 4)

$15$

Incorrect

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