First slide

Binomial theorem for positive integral Index

Question

The coefficients of three consecutive terms of $(1 + x)^{n + 5}$ are in the ratio $5 : 10 : 14$ . Then, $n =$

Moderate

A

5
B

7
C

6
D

8

Solution

Let , $r^{t h}, (r + 1)^{t h}$ and $(r + 2)^{t h}$ be three consecutive terms in the expansion of $(1 + x)^{n + 5}$
It is given that
$^{n + 5} C_{r - 1} :^{n + 5} C_{r} :^{n + 5} C_{r + 1} = 5 : 10 : 14$
$\Rightarrow \frac{^{n + 5} C_{r}}{^{1 + 3} C_{r - 1}} = \frac{10}{5}$ and $\frac{^{n + 5} C_{r + 1}}{^{n + 5} C_{r}} = \frac{14}{10}$
$\Rightarrow \frac{n + 5 - r + 1}{r} = 2$ and $\frac{n + 5 - r}{r + 1} = \frac{7}{5}$
$\Rightarrow n - 3 r + 6 = 0$ and $5 n - 12 r + 18 = 0$
$\Rightarrow n = 6, r = 4$

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