Binomial Expansion

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If the coefficient of x⁷ and x⁸ in (2 + x/3)ⁿ are equal, then n equals

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1. **General Coefficient**: In $(a+bx)^n$, the coefficient of $x^r$ is $\binom{n}{r} a^{n-r} (b)^r$.
   - Here $a=2, b=1/3$.
2. **Equal Coefficients Condition**:
   - $	ext{Coeff of } x^7 = 	ext{Coeff of } x^8$
   - $\binom{n}{7} 2^{n-7} (\frac{1}{3})^7 = \binom{n}{8} 2^{n-8} (\frac{1}{3})^8$
3. **Division**:
   - $\frac{\binom{n}{7}}{\binom{n}{8}} \cdot \frac{2^{n-7}}{2^{n-8}} = \frac{(1/3)^8}{(1/3)^7}$
   - $\frac{8}{n-7} \cdot 2 = \frac{1}{3}$
4. **Equation**: $16 	imes 3 = n - 7 \implies 48 = n - 7 \implies n = 55$.

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