Elaborate Solution for Mean of Coefficients

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1. Key Concept: The Binomial Theorem

The core concept for solving this problem is the Binomial Theorem, which provides a formula for expanding algebraic expressions of the form $(a+b)^n$. The general term (or $(r+1)$-th term) in the binomial expansion of $(a+b)^n$ is given by: $T_{r+1} = {}^n C_r a^{n-r} b^r$ where ${}^n C_r = \frac{n!}{r!(n-r)!}$ is the binomial coefficient, representing the number of ways to choose $r$ items from a set of $n$ items.

In our problem, we have the expansion of $(2+x)^9$. Comparing this with $(a+b)^n$, we have $a=2$, $b=x$, and $n=9$. Therefore, the general term for $(2+x)^9$ is: $T_{r+1} = {}^9 C_r (2)^{9-r} (x)^r$ The coefficient of $x^r$ in this expansion is ${}^9 C_r 2^{9-r}$.

2. Step-by-Step Working

Step 1: Identify the Coefficients of the Required Terms

We need to find the mean of the coefficients of $x, x^2, \ldots, x^7$. This means we need the coefficients for $r=1, 2, \ldots, 7$.

• Coefficient of $x^1$ (when $r=1$): ${}^9 C_1 2^{9-1} = {}^9 C_1 2^8$
• Coefficient of $x^2$ (when $r=2$): ${}^9 C_2 2^{9-2} = {}^9 C_2 2^7$
• Coefficient of $x^3$ (when $r=3$): ${}^9 C_3 2^{9-3} = {}^9 C_3 2^6$
• ...
• Coefficient of $x^7$ (when $r=7$): ${}^9 C_7 2^{9-7} = {}^9 C_7 2^2$

Let $C_r$ denote the coefficient of $x^r$, so $C_r = {}^9 C_r 2^{9-r}$. We are interested in the coefficients $C_1, C_2, \ldots, C_7$.

Step 2: Formulate the Mean Expression

The mean of these 7 coefficients is their sum divided by the count of coefficients, which is 7. $\text{Mean} = \frac{C_1 + C_2 + C_3 + C_4 + C_5 + C_6 + C_7}{7} = \frac{\sum_{r=1}^{7} {}^9 C_r 2^{9-r}}{7}$

Step 3: Utilize the Full Binomial Expansion for Summation

Calculating each of these 7 coefficients individually and then summing them would be very tedious and prone to error. A more elegant approach involves using the full binomial expansion.

Recall the full expansion of $(2+x)^9$: $(2+x)^9 = {}^9 C_0 2^9 x^0 + {}^9 C_1 2^8 x^1 + {}^9 C_2 2^7 x^2 + \ldots + {}^9 C_7 2^2 x^7 + {}^9 C_8 2^1 x^8 + {}^9 C_9 2^0 x^9$

The sum of all coefficients in the expansion of $(a+b)^n$ can be found by substituting $x=1$ (if $b=x$) or simply summing the numerical parts of each term. In our case, the sum of all coefficients for $(2+x)^9$ is equivalent to evaluating the expression when $x=1$: $(2+1)^9 = 3^9$ This sum $3^9$ includes the coefficients of $x^0, x^1, \ldots, x^9$. Let $S_{total}$ be this sum. $S_{total} = \sum_{r=0}^{9} {}^9 C_r 2^{9-r} = {}^9 C_0 2^9 + \left( \sum_{r=1}^{7} {}^9 C_r 2^{9-r} \right) + {}^9 C_8 2^1 + {}^9 C_9 2^0$ Let $S_{required} = \sum_{r=1}^{7} {}^9 C_r 2^{9-r}$ be the sum we need to calculate for the mean. We can express $S_{required}$ as: $S_{required} = S_{total} - \left( {}^9 C_0 2^9 + {}^9 C_8 2^1 + {}^9 C_9 2^0 \right)$

Step 4: Calculate the Values

First, calculate $S_{total}$: $S_{total} = 3^9 = 19683$

Next, calculate the coefficients of the terms we need to subtract:

• For $r=0$: Coefficient of $x^0$ is ${}^9 C_0 2^9$.
  • ${}^9 C_0 = 1$
  • $2^9 = 512$
  • So, ${}^9 C_0 2^9 = 1 \times 512 = 512$
• For $r=8$: Coefficient of $x^8$ is ${}^9 C_8 2^1$.
  • Using the property ${}^n C_r = {}^n C_{n-r}$, we have ${}^9 C_8 = {}^9 C_{9-8} = {}^9 C_1 = 9$.
  • $2^1 = 2$
  • So, ${}^9 C_8 2^1 = 9 \times 2 = 18$
• For $r=9$: Coefficient of $x^9$ is ${}^9 C_9 2^0$.
  • ${}^9 C_9 = 1$
  • $2^0 = 1$
  • So, ${}^9 C_9 2^0 = 1 \times 1 = 1$

Now, substitute these values back into the equation for $S_{required}$: $S_{required} = 19683 - (512 + 18 + 1)$ $S_{required} = 19683 - 531$ $S_{required} = 19152$

Step 5: Calculate the Mean

Finally, divide the sum of the required coefficients by 7 (the number of terms): $\text{Mean} = \frac{19152}{7} = 2736$

3. Tips and Common Mistakes to Avoid

• Counting Terms Correctly: Always be careful about the range of terms. In this problem, we needed coefficients from $x^1$ to $x^7$, which is $7-1+1 = 7$ terms. If $x^0$ were included, it would be 8 terms.
• Binomial Coefficient Properties: Remember properties like ${}^n C_r = {}^n C_{n-r}$ to simplify calculations (e.g., ${}^9 C_8 = {}^9 C_1$).
• Full Expansion Sum: The sum of all coefficients in $(a+b)^n$ is $(a+b)^n$ evaluated at $x=1$ (or $b=1$ if $a$ is the variable). In this case, it's $(2+1)^9$.
• Careful with Arithmetic: The calculations involve relatively large numbers ($3^9$), so use a calculator or perform arithmetic carefully.

4. Summary

To find the mean of a specific range of coefficients in a binomial expansion:

1. Identify the general term and its coefficient.
2. Determine the range of $r$ values for the required coefficients.
3. Calculate the sum of all coefficients in the expansion using the property $(a+b)^n$.
4. Subtract the coefficients of the terms not included in the required range from the total sum.
5. Divide the resulting sum by the count of the required coefficients to find the mean. By using the full binomial expansion identity, we avoid lengthy individual calculations and arrive at the solution efficiently. The mean of the coefficients of $x, x^2, \ldots, x^7$ in the binomial expansion of $(2+x)^9$ is $2736$.