Key Concept: Binomial Theorem and General Term

The Binomial Theorem provides a formula for the algebraic expansion of powers of a binomial. For any positive integer $n$, the expansion of $(a+b)^n$ is given by: $(a+b)^n = \sum_{r=0}^{n} {}^nC_r a^{n-r} b^r$ The $(r+1)$-th term in the binomial expansion of $(a+b)^n$ is known as the general term, denoted by $T_{r+1}$, and is given by: $T_{r+1} = {}^nC_r a^{n-r} b^r$ In this problem, we need to find the coefficients of specific powers of $x$. The coefficient of $x^k$ is the part of the term $T_{r+1}$ that does not include $x$.

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Step-by-step Working with Explanations

1. Identify the General Term for the Given Expansion

The given expression is ${\left( {2 + {x \over 3}} \right)^n}$. Comparing this with $(a+b)^n$, we identify $a=2$ and $b=\frac{x}{3}$.

Using the general term formula $T_{r+1} = {}^nC_r a^{n-r} b^r$, we substitute the values of $a$ and $b$: $T_{r+1} = {}^nC_r (2)^{n-r} \left(\frac{x}{3}\right)^r$ To separate the coefficient from the variable $x$, we can rewrite the term as: $T_{r+1} = {}^nC_r (2)^{n-r} \left(\frac{1}{3}\right)^r x^r$

2. Determine the Coefficients of $x^7$ and $x^8$

• For the coefficient of $x^7$: We need the power of $x$ in the general term to be $7$. From $x^r$, this implies $r=7$. Substituting $r=7$ into the general term, the coefficient of $x^7$ is: $C_7 = {}^nC_7 (2)^{n-7} \left(\frac{1}{3}\right)^7 = {}^nC_7 2^{n-7} \frac{1}{3^7}$

• For the coefficient of $x^8$: Similarly, for $x^8$, we need $r=8$. Substituting $r=8$ into the general term, the coefficient of $x^8$ is: $C_8 = {}^nC_8 (2)^{n-8} \left(\frac{1}{3}\right)^8 = {}^nC_8 2^{n-8} \frac{1}{3^8}$

3. Equate the Coefficients

The problem states that the coefficient of $x^7$ and $x^8$ are equal. Therefore, we set $C_7 = C_8$: ${}^nC_7 2^{n-7} \frac{1}{3^7} = {}^nC_8 2^{n-8} \frac{1}{3^8}$

4. Simplify the Equation and Solve for $n$

To simplify, we can rearrange the terms by bringing similar components to one side. We'll divide both sides by common factors. Divide both sides by ${}^nC_7$, $2^{n-8}$, and $\frac{1}{3^7}$: $\frac{{}^nC_8}{{}^nC_7} \cdot \frac{2^{n-8}}{2^{n-7}} \cdot \frac{1/3^8}{1/3^7} = 1 \quad \text{(This step is incorrect, as it would imply that the product of the terms is 1, not that they are equal. Let's do it differently)}$

Let's directly manipulate the equation: ${}^nC_7 2^{n-7} \frac{1}{3^7} = {}^nC_8 2^{n-8} \frac{1}{3^8}$ Divide both sides by $2^{n-8}$ and $\frac{1}{3^7}$: ${}^nC_7 \cdot 2^{n-7 - (n-8)} = {}^nC_8 \cdot \frac{1}{3^8 \cdot 3^{-7}}$ ${}^nC_7 \cdot 2^1 = {}^nC_8 \cdot \frac{1}{3^1}$ $2 \cdot {}^nC_7 = \frac{1}{3} \cdot {}^nC_8$ Now, we use the identity for binomial coefficients: $\frac{{}^nC_r}{{}^nC_{r-1}} = \frac{n-r+1}{r}$. Here, we have ${}^nC_8$ and ${}^nC_7$. Let $r=8$, then $r-1=7$. So, $\frac{{}^nC_8}{{}^nC_7} = \frac{n-8+1}{8} = \frac{n-7}{8}$.

Substitute this into our equation: $2 = \frac{1}{3} \cdot \frac{{}^nC_8}{{}^nC_7}$ $2 = \frac{1}{3} \cdot \frac{n-7}{8}$ Now, solve for $n$: $2 = \frac{n-7}{24}$ Multiply both sides by $24$: $2 \times 24 = n-7$ $48 = n-7$ Add $7$ to both sides: $n = 48 + 7$ $n = 55$

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Tips and Common Mistakes to Avoid

• Careful with 'a' and 'b': Ensure you correctly identify $a$ and $b$ in the binomial expression $(a+b)^n$. In this case, $b = \frac{x}{3}$, not just $x$. The $\frac{1}{3}$ factor is crucial.
• Powers of the constant term: Don't forget to include the powers of the constant term (here, $2^{n-r}$) in the coefficient calculation.
• Binomial Coefficient Identity: Remember and correctly apply the ratio identity for binomial coefficients: $\frac{{}^nC_r}{{}^nC_{r-1}} = \frac{n-r+1}{r}$. This significantly simplifies the algebraic manipulation compared to expanding factorials.
• Algebraic Simplification: Be meticulous with algebraic simplification, especially when dealing with exponents and fractions. A common mistake is to miscalculate $2^{n-7} / 2^{n-8}$ or $3^{-7} / 3^{-8}$.

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Summary

By correctly identifying the general term of the binomial expansion, extracting the coefficients of $x^7$ and $x^8$, and then equating them, we formed an equation involving $n$. Utilizing the property of binomial coefficients $\frac{{}^nC_8}{{}^nC_7} = \frac{n-7}{8}$ allowed for straightforward algebraic manipulation to find the value of $n$. The final answer is $n=55$. This problem highlights the importance of understanding the general term formula and binomial coefficient identities.