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

  1. $2$
  2. $\frac{7}{4}$
  3. $\frac{3}{2}$
  4. $\frac{4}{3}$
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1 Answer

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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) $\dfrac{4}{3}$  

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