Q110 / 170 · Manipal (MET) 2018 · Mathematics

If the coefficients of the $r$ th term and the $(r+1)$ th term in the expansion of $(1+x)^{20}$ are in the ratio $1: 2$, then $r$ is equal to

  • A
    6
  • Correct answer
    B
    7
  • C
    8
  • D
    9

Solution

$T_r=T_{(r-1)+1}={ }^{20} C_{r-1} \cdot x^{r-1}$ $\therefore$ Coefficient of the $r$ th term $={ }^{20} C_{r-1}$ and coefficient of the $(r+1)$ th term $={ }^{20} C_r$ $\therefore \quad \frac{{ }^{20} C_{r-1}}{{ }^{20} C_r}=\frac{1}{2}$ $\begin{array}{lr}\Rightarrow & \frac{r}{21-r}=\frac{1}{2} \\ \Rightarrow & r=7\end{array}$

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

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