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CAT 2017 Slot 2 QA Question & Solution

AlgebraMedium

Question

An infinite geometric progression $a_1,a_2,...$ has the property that $a_n= 3(a_{n+1}+ a_{n+2} + ...)$ for every n $\geq$ 1. If the sum $a_1+a_2+a_3...+=32$, then $a_5$ is

Options

1/32
2/32
3/32
4/32

Solution

Let the common ratio of the G.P. be r.
Hence we have $a_n= 3(a_{n+1}+ a_{n+2} + ...)$

The sum up to infinity of GP is given by $\frac{a}{1-r}$ where a here is $a_{n+1}$

=> $a_n= 3(\frac{a_{n+1}}{1-r})$
=> $a_n= 3(\frac{a_{n}\times r}{1-r})$
=> $r = \frac{1}{4}$
Now, $a_1+a_2+a_2...+=32$
=> $\frac{a_1}{1-r} = 32$
=> $\frac{a_1}{3/4} = 32$
=> $a_1 = 24$

$a_5 = a_1 \times r^4$
$a_5 = 24 \times (1/4)^4 = \frac{3}{32}$