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Q.
If the coefficients of the $r^{th}$ term and $(r + 1)^{th}$ term in the expansion of $(1 + x)^{20}$ are in the ratio $1 : 2$, then $r$ is equal to
Binomial Theorem
Solution:
$r^{th}$ term in the expansion of $(1 + x)^{20}$ is $^{20}C_{r- 1} \,(x)^{(r-1)}$
$(r + 1)^{th}$ term in the expansion of $(1 + x)^{20}$ is $^{20}C_r(x)^r$
Now, $\frac{^{20}C_{r-1}}{^{20}C_{r}} = \frac{1}{2}$
$\Rightarrow \frac{r}{21-r} = \frac{1}{2}$
$\Rightarrow 2r = 21 -r$
$\Rightarrow r = 7$.