Browse Questions

The two successive terms in the expansion of $(1+x)^{24}$ whose coefficients are in the ratio $1 : 4$ are

$\begin{array}{1 1}(A)\;3^{rd}\;and\;4^{th}\\(B)\;4^{th}\;and\;5^{th}\\(C)\;5^{th}\;and\;6^{th}\\(D)\;6^{th}\;and\;7^{th}\end{array}$

Toolbox:
• $T_{r+1}=nC_ra^{n-r}b^r$
The two successive terms are $nC_r,nC_{r+1}$
$\Rightarrow$ The ratio given is $1 : 4$
$\Rightarrow \large\frac{24C_r}{24C_{r+1}}=\frac{1}{4}$
$\Rightarrow \large\frac{r+1}{24-r}=\frac{1}{4}$
$\Rightarrow 24-r=4(r+1)$
$\Rightarrow 24-r=4r+4$
$\Rightarrow -4r-r=4-24$
$\Rightarrow -5r=-20$
$\Rightarrow r=4$
Hence the two consecutive terms are $4^{th}$ and $5^{th}$
Hence (B) is the correct answer.