First slide

Geometric progression

Question

In a geometric progression the ratio of the sum of the first 5 terms to the sum of their reciprocals is 49 and sum of the first and the third term is 35.The fifth term of the G.P. is

Moderate

A

7
B

7/2
C

7/4
D

7/8

Solution

Let five terms in G.P. be $\frac{a}{r^{2}}, \frac{a}{r}, a, a r, a r^{2}$
Then $\frac{a (r^{- 2} + r^{- 1} + 1 + r + r^{2})}{(1 / a) (r^{2} + r + 1 + r^{- 1} + r^{- 2})} = 49$
$\Rightarrow a^{2} = 49 \Rightarrow a = \pm 7 .$
Also, $\frac{a}{r^{2}} + a = 35$
Therefore $a = - 7$ is not possible
Thus, $a = 7$ and $a / r^{2} = 28 .$
Now, fifth term $= a r^{2} = a (\frac{a}{28}) = \frac{7}{4}$

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