If the coefficients of $r$ th, $(r+1)$ th and $(r+2)$ th terms in the binomial expansion of $(1+$ $y)^m$ are…

If the coefficients of $r$ th, $(r+1)$ th and $(r+2)$ th terms in the binomial expansion of $(1+$ $y)^m$ are in A.P., then $m$ and $r$ satisfy the equation
  1. $m^2-m(4 r-1)+4 r^2-2=0$
  2. $m^2-m(4 r+1)+4 r^2+2=0$
  3. $m^2-m(4 r+1)+4 r^2-2=0$
  4. $m^2-m(4 r-1)+4 r^2+2=0$

Solution

Given ${ }^m C,{ }^m C,{ }^m C_{r+1}$ are in A.P. $ \begin{aligned} & 2{ }^m C_r={ }^m C_{r-1}+{ }^m C_{r+1} \\ & \Rightarrow 2=\frac{{ }^m C_{r-1}}{{ }^m C_r}+\frac{{ }^m C_{r+1}}{{ }^m C_r} \\ & =\frac{r}{m-r+1}+\frac{m-r}{r+1} \\ & \Rightarrow m^2-m(4 r+1)+4 r^2-2=0 \end{aligned} $

Asked in: JEE Main 2005

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