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  4. In a geometric progression, if the ratio of the sum of first 5 terms to the sum of their reciprocals is 49, and the sum of the first and the third term is 35. Then the first term of this geometric progression is :

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Q. In a geometric progression, if the ratio of the sum of first $5$ terms to the sum of their reciprocals is $49$, and the sum of the first and the third term is $35$. Then the first term of this geometric progression is :

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Solution:

Let first term is a & $C.R = r$
given $\frac{\left(a + ar + ar^{2} + ar^{3} + ar^{4}\right)}{\left(\frac{1}{a}+\frac{1}{ar}+\frac{1}{ar^{2}}+\frac{1}{ar^{3}}+\frac{1}{ar^{4}}\right)} = 49$
$a^{2} r^{4} = 49 \Rightarrow ar^{2} =7,-7$
also given that $a + ar^{2} = 35$
if $ar^{2} = 7 \Rightarrow a = 35 - 7 = 28$
if $ar^{2} = - 7 \Rightarrow a = 35 + 7 = 42$
but if $a = 42$ then $r^{2} = -\frac{7}{42}$
which is not possible so
$a = 28$