GATE ECE 2018 | GA | Question: 4

Question

What is the value of $1+\frac{1}{4}+\frac{1}{16}+\frac{1}{256}+ \cdots?$

• A. $2$
• B. $\frac{7}{4}$
• C. $\frac{3}{2}$
• D. $\frac{4}{3}$

Answer 1

It is an infinite G.P. with first term $a=1$ and common ratio $r = \dfrac{1}{4}$.

Sum of infinite G.P with $|r| < 1$ is

$S_{\infty}=\dfrac{a}{1-r}$

$S_{\infty}=\dfrac{1}{1-\dfrac{1}{4}}$

$S_{\infty}=\dfrac{4}{3}$

Hence option d) is correct

Answer 2

1 + $\dfrac{1}{4}$ + $\dfrac{1}{16}$ + $\dfrac{1}{64}$ + $\dfrac{1}{256}$ + ....

By seeing the above series we can identify the series as infinite G.P. series

Here 1st term i.e. a = 1 & common ratio i.e r = $\dfrac{1}{4}$

∴ The sum of the above infinite G.P series will be $\dfrac{1}{1 - \dfrac{1}{4} }$  = $\dfrac{4}{3}$  

∴ The answer is option D) $\dfrac{4}{3}$  

Answer 3

Answer: D

Answer 4

Using gp sum of infinite  a /1-r so answer is 4/3

Answer 5

IT IS IN GP

SO IN GP INFINITE SERIES IS

S∞​=a/(1−r)

     =1/1-¼

​      =4/3