Let the coefficients of three consecutive terms in the binomial expansion of be in the ratio 2:5:8.Then the coefficient of the term, which is in the middle of these three terms, is _________ . [2023]

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Solution

Q.
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 _________ . [2023]

Ans.
(1120)

$Coefficient of {(r + 1)}^{th} term is,$

$t_{r + 1} = {}^{n}C_{r} {(2 x)}^{r}$

Given consecutive term ratio as,

$\frac{{}^{n}C_{r - 1} {(2)}^{r - 1}}{{}^{n}C_{r} {(2)}^{r}} = \frac{2}{5} \Rightarrow \frac{n! / (r - 1)! (n - r + 1)!}{n! (2) / r! (n - r)!} = \frac{2}{5}$

$\Rightarrow \frac{r}{n - r + 1} = \frac{4}{5} \Rightarrow 5 r = 4 n - 4 r + 4 \Rightarrow 9 r = 4 (n + 1) ...(i)$

Again, consecutive term ratio are,

$\frac{{}^{n}C_{r} {(2)}^{r}}{{}^{n}C_{r + 1} {(2)}^{r + 1}} = \frac{5}{8} \Rightarrow \frac{n! / r! (n - r)!}{n! / (r + 1)! (n - r - 1)!} = \frac{5}{4}$

$\Rightarrow \frac{r + 1}{n - r} = \frac{5}{4} \Rightarrow 5 n - 4 = 9 r ...(ii)$

$From (i) and (ii): n = 8, r = 4$

$So, coefficient of middle term is,$

${}^{8}C_{4} 2^{4} = 16 \times \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = 1120$

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