Binomial Theorem Ques 36
The coefficients of three consecutive terms of $(1+x)^{n+5}$ are in the ratio $5: 10: 14$. Then, $n$ is equal to
(2013 Adv.)
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Answer:
Correct Answer: 36.$(n=6)$
Solution:
Formula:
Properties of binomial theorem:
- Let the three consecutive terms in $(1+x)^{n+5}$ be $t_{r}, t_{r+1}, t_{r+2}$ having coefficients
${ }^{n+5} C_{r-1},{ }^{n+5} C_{r},{ }^{n+5} C_{r+1}$.
Given, ${ }^{n+5} C_{r-1}:{ }^{n+5} C_{r}:{ }^{n+5} C_{r+1}=5: 10: 14$
$\therefore \quad \frac{{ }^{n+5} C_{r}}{{ }^{n+5} C_{r-1}}=\frac{10}{5}$ and $\frac{{ }^{n+5} C_{r+1}}{{ }^{n+5} C_{r}}=\frac{14}{10}$
$\Rightarrow \quad \frac{n+5-(r-1)}{r}=2$ and $\frac{n-r+5}{r+1}=\frac{7}{5}$
$\Rightarrow \quad n-r+6=2 r$ and $5 n-5 r+25=7 r+7$
$\Rightarrow \quad n+6=3 r$ and $5 n+18=12 r$
$\therefore \quad \frac{n+6}{3}=\frac{5 n+18}{12}$
$\Rightarrow \quad 4 n+24=5 n+18 \Rightarrow \quad n=6$
BETA
