Question

Consider an infinite geometric series with first term $a$and common ratio $r.$ If its sum is $4$ and the second term is $\frac{3}{4}$, then

• A. $a=\frac{4}{7},r=\frac{3}{7}$
• B. $a=2,r=\frac{3}{8}$
• C. $a=\frac{3}{2},r=\frac{1}{2}$
• D. $a=3,r=\frac{1}{4}$

Answer 1

The correct option is D

$a=3,r=\frac{1}{4}$

Finding the value:

Now, Sum of infinite series $=\frac{a}{1-r}$$=4$ -----(1)

Second Term$=ar=\frac{3}{4}$ ------(2)

Substitute equation (2) in equation (1),we have

$a+4\times \left(\frac{3}{4a}\right)=4$

$\Rightarrow$ $a^2-4a+3=0$

$\Rightarrow$ $(a-3)(a-1)=0$

$\Rightarrow$ $a=3ora=1$

$$\begin{gathered} Whena=3,r=\frac{1}{4} \\ Whena=1,r=\frac{3}{4} \end{gathered}$$

Hence, Option(D) is correct.