Question

If the coefficients of $x^7$ and $x^8$ in ${(2+\frac{x}{3})}^n$ are equal, then $x$ is?

• A. $56$
• B. $55$
• C. $45$
• D. $15$

Answer 1

Answer: (B)

$55$

${(2+\frac{x}{3})}^x$

$T_{r+1}={}^n{C}_r{2}^{n-r}(x/3{)}^r$

$T_8={}^n{C}_7\frac{2^{n-7}}{3^7}$ where $r=7$

$T_9={}^n{C}_8\frac{2^{n-8}}{3^8}r=8$

$T_8=T_9$

${}^n{C}_7\frac{2^{n-7}}{3^7}={}^n{C}_8\frac{2^{n-8}}{3^8}$

$\frac{n!}{7!(n-7)!}\frac{2^{n-7}}{3^7}=\frac{n!}{8!(n-8)!}\frac{2^{n-8}}{3^8}$

$n-7=48$

$n=55$