First slide

Binomial theorem for positive integral Index

Question

If the coefficients of three consecutive terms in the expansion of ${(1 + x)}^{n}$ are in the ratio $1 : 7 : 42$ , then value of $n$ is

Moderate

A

$45$
B

$55$
C

$65$
D

$75$

Solution

Let three consecutive terms be $(r + 1)$ th, $(r + 2)$ th and $(r + 3)$ th. Then
$\frac{^{n} C_{r}}{^{n} C_{r + 1}} = \frac{1}{7} \Rightarrow \frac{r + 1}{n - r} = \frac{1}{7} \Rightarrow n - 8 r = 7$
and $\frac{^{n} C_{r + 1}}{^{n} C_{r + 2}} = \frac{7}{42} = \frac{1}{6} \Rightarrow \frac{r + 2}{n - r - 1} = \frac{1}{6}$
$\Rightarrow n - 7 r = 13$ |
From $(1)$ and $(2), n = 55 .$

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