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Binomial theorem for positive integral Index

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Question

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

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Solution

$T h e g i v e n b i n o m i a l e x p a n s i o n i s {(2 + \frac{x}{3})}^{n} General term in above expansion is T_{r + 1} = n C_{r} (2^{n - r}) {(\frac{x}{3})}^{r} = {}^{n}C_{r} (2)^{n - r} {(\frac{1}{3})}^{r} x^{r} Coefficient of x^{7} = Coefficient of x^{8} {}^{n}C_{7} 2^{n - 7} {(\frac{1}{3})}^{7} = {}^{n}C_{8} (2)^{n - 8} {(\frac{1}{3})}^{8} \Rightarrow {}^{n}C_{7} \times 2 = {}^{n}C_{8} \cdot \frac{1}{3} \Rightarrow \begin{array}{l} \Rightarrow \frac{n!}{7! (n - 7)!} \cdot 2 = \frac{n!}{(n - 8)! 8!} \cdot \frac{1}{3} \\ \frac{2}{n - 7} = \frac{1}{24} \\ \Rightarrow n - 7 = 48 \\ \Rightarrow n = 55 \end{array}$

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