Question

find value of n in above question​

Answer 1

Answer:

The value of n is 55

Step-by-step explanation:

$\text{In the expansion of }(2+\frac{x}{3})^n,\text{ if the coefficients of }x^7\text{ and }x^8\text{ are equal}$

$\textbf{Concept used:}$

$\text{The general term in the expansion of }(a+b)^n\text{ is}$

$\boxed{\bf\:T_{r+1}=^nC_r\:a^{n-r}\:b^r}$

$\text{in the expansion of }(2+\frac{x}{3})^n\:\text{ the general term is}$

$\:T_{r+1}=^nC_r\:a^{n-r}\:b^r$

$\:T_{r+1}=^nC_r\:2^{n-r}\:(\frac{x}{3})^r$

$\:T_{r+1}=^nC_r\:\frac{2^n}{2^r}\:(\frac{x^r}{3^r})$

$\:T_{r+1}=^nC_r\:(\frac{2^n}{6^r})\:x^r$

$\text{Put r=7}$

$\:T_{8}=^nC_7\:(\frac{2^n}{6^7})\:x^7$

$\text{Put r=8}$

$\:T_{9}=^nC_8\:(\frac{2^n}{6^8})\:x^8$

$\text{ since the coefficients of }x^7\text{ and }x^8\text{ are equal}$

$\implies\:^nC_7\:(\frac{2^n}{6^7})=^nC_8\:(\frac{2^n}{6^8})$

$\implies\:^nC_7\:(\frac{1}{6^7})=^nC_8\:(\frac{1}{6^8})$

$\implies\:^nC_7=^nC_8\:(\frac{1}{6})$

$\implies\:6(^nC_7)=^nC_8$

$\implies\:6=\dfrac{^nC_8}{^nC_7}$

Using

$\boxed{\bf\frac{^nC_r}{^nC_{r-1}}=\frac{n-r+1}{r}}$

$\implies\:6=\dfrac{n-8+1}{8}$

$\implies\:6=\dfrac{n-7}{8}$

$\implies\:48=n-7$

$\implies\:\boxed{n=55}$