Q70 / 125 · BITSAT 2021 · Mathematics

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
    $a=\frac{3}{2}, r=\frac{1}{2}$
  • Correct answer
    D
    $a=3, r=\frac{1}{4}$

Solution

Since, sum $=4$ and second term $=\frac{3}{4}$ $\Rightarrow \frac{a}{1-r}=4$, and $a r=\frac{3}{4}$ $\Rightarrow \frac{a}{1-\frac{3}{4 a}}=4$ $\Rightarrow(a-1)(a-3)=0$ $\Rightarrow a=1$ or $a=3$ $\begin{array}{l} \Rightarrow \frac{a}{1-\frac{3}{4 a}}=4 \\ \Rightarrow(a-1)(a-3)=0 \\ \Rightarrow a=1 \text { or } a=3 \end{array}$

Hence, the correct answer is option: 'D'.

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