Question

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 :

• A. 7
• B. 21
• C. 28
• D. 42

Answer 1

Answer: (C)

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