Binomial Theorem Question 10
Question 10 - 29 January - Shift 1
Let the coefficients of three consecutive terms in the binomial expansion of $(1+2 x)^{n}$ be in the ratio $2: 5: 8$. Then the coefficient of the term, which is in the middle of these three terms, is_______
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Answer: 1120
Solution:
Formula: General term of Bionomial Cofficient
$ t _{r+1}={ }^{n} C_r(2 x)^{r} $
$\Rightarrow \frac{{ }^{n} C _{r-1}(2)^{r-1}}{{ }^{n} C_r(2)^{r}}=\frac{2}{5}$
$\Rightarrow \frac{\frac{n !}{(r-1) !(n-r+1) !}}{\frac{n !(2)}{r !(n-r) !}}=\frac{2}{5}$
$\Rightarrow \frac{r}{n-r+1}=\frac{4}{5} \Rightarrow 5 r=4 n-4 r+4$
$\Rightarrow 9 r=4(n+1)\ldots (1) $
$\Rightarrow \frac{{ }^{n} C_r(2)^{r}}{{ }^{n} C _{r+1}(2)^{r+1}}=\frac{5}{8}$
$ \begin{aligned} & \Rightarrow \frac{\frac{n !}{r !(n-r) !}}{n !}=\frac{5}{4} \Rightarrow \frac{r+1}{n-r}=\frac{5}{4} \\ & \overline{(r+1) !(n-r-1) !} \\ & \Rightarrow 4 r+4=5 n-5 r \Rightarrow 5 n-4=9 r \ldots (2) \end{aligned} $
From (1) and (2)
$\Rightarrow 4 n+4=5 n-4 \Rightarrow n=8$
(1) $\Rightarrow r=4$
so, coefficient of middle term is
