Question

The coefficient of three consecutive of ${(1+x)}^{n+5}$ are in the ratio $5:10:14$. Then, $n$ is equal to (answer lies between $0$ to $9$)

• A. $1$
• B. $3$
• C. $4$
• D. $6$

Answer 1

The correct option is D

$6$

Explanation for the correct option:

Finding value of $n$ for the given expression:

Given expression is ${(1+x)}^{n+5}$

Now, we have

$$\begin{gathered} T_{r-1}:T_r:T_{r+1}=5:10:14 \\ \frac{T_{r+1}}{T_r}=\frac{n+5-r+1}{r},\frac{T_r}{T_{r-1}}=\frac{n+5-r+2}{r-1} \\ \Rightarrow \frac{T_{r+1}}{T_r}=\frac{14}{10},\frac{T_r}{T_{r-1}}=\frac{10}{5} \end{gathered}$$ $[\text{∵Ratio of binomial coeff}.\frac{T_{r+1}}{T_r}=\frac{n-r+1}{r}]$

Now from above,

$\Rightarrow \frac{n+6-r}{r}=\frac{14}{10},\frac{n+7-r}{r-1}=\frac{10}{5}$

So, solving for $n$ we get

$$\begin{gathered} \Rightarrow 5n+30-5r=7r;n+7-r=2r-2 \\ \Rightarrow 5n+30=12r;n+9=3r \\ \text{Solve for n} \\ \Rightarrow 5n+30=4n+36 \\ \Rightarrow n=6 \end{gathered}$$

Hence, Option(D) is correct answer