If the co-efficient of x^7 and x^8 in the expansion of (2+ x3)^n are equal, then the value…

Question

If the co-efficient of $x^7$ and $x^8$ in the expansion of $\left(2+\dfrac x3\right)^n$ are equal, then the value of $n$ is equal to ..........

Answer: 55

Accepted Answer

$$\binom n7 2^{n-7}\left(\frac13\right)^7=\binom n8 2^{n-8}\left(\frac13\right)^8$$

$$\frac{n!}{7!(n-7)!}\times2=\frac{n!}{8!(n-8)!}\times\frac13$$

$$\frac{2}{n-7}=\frac{1}{8\times3}\quad\Rightarrow\quad\frac{2}{n-7}=\frac{1}{24}\quad\Rightarrow\quad n-7=48\quad\Rightarrow\quad n=55$$

Solutions · 1