The power of x which has the greatest coefficient inthe expansion of `(1+x/2)^10`

Options

A. 2
B. 3
C. 4
D. 5

Explanation

To find the power of \( x \) which has the greatest coefficient in the expansion of \( \left(1 + \frac{x}{2}\right)^{10} \), we can use the Binomial Theorem.

### Step-by-Step Solution:

1. **Identify the General Term**:
The general term \( T_{r+1} \) in the expansion of \( \left(1 + \frac{x}{2}\right)^{10} \) is given by:
\[ T_{r+1} = \binom{10}{r} \left(\frac{x}{2}\right)^r = \binom{10}{r} \frac{x^r}{2^r} \]

2. **Find the Coefficient of \( x^r \)**:
The coefficient of \( x^r \) in the term \( T_{r+1} \) is:
\[ C_r = \binom{10}{r} \frac{1}{2^r} \]

3. **Set Up the Ratio of Consecutive Coefficients**:
To find the maximum coefficient, we can compare consecutive coefficients \( C_r \) and \( C_{r+1} \):
\[ \frac{C_{r+1}}{C_r} = \frac{\binom{10}{r+1} \frac{1}{2^{r+1}}}{\binom{10}{r} \frac{1}{2^r}} = \frac{\binom{10}{r+1}}{\binom{10}{r}} \cdot \frac{1}{2} = \frac{10 - r}{r + 1} \cdot \frac{1}{2} \]

4. **Determine When the Ratio is Greater Than 1**:
We want to find \( r \) such that:
\[ \frac{10 - r}{2(r + 1)} > 1 \]
Simplifying this inequality:
\[ 10 - r > 2(r + 1) \]
\[ 10 - r > 2r + 2 \]
\[ 10 - 2 > 3r \]
\[ 8 > 3r \quad \Rightarrow \quad r < \frac{8}{3} \approx 2.67 \]

5. **Determine When the Ratio is Less Than 1**:
Now we check the next ratio:
\[ \frac{C_r}{C_{r-1}} < 1 \]
This gives:
\[ \frac{r}{10 - r + 1} < 2 \quad \Rightarrow \quad r < \frac{10 - r + 1}{2} \]
Simplifying:
\[ 2r < 11 - r \quad \Rightarrow \quad 3r < 11 \quad \Rightarrow \quad r < \frac{11}{3} \approx 3.67 \]

6. **Finding Integer Values**:
From the inequalities \( r < \frac{8}{3} \) and \( r < \frac{11}{3} \), the integer values of \( r \) that satisfy both conditions are \( r = 0, 1, 2, 3 \).

7. **Finding the Maximum Coefficient**:
We can evaluate \( C_r \) for \( r = 0, 1, 2, 3 \):
- For \( r = 0 \): \( C_0 = \binom{10}{0} \cdot \frac{1}{2^0} = 1 \)
- For \( r = 1 \): \( C_1 = \binom{10}{1} \cdot \frac{1}{2^1} = 5 \)
- For \( r = 2 \): \( C_2 = \binom{10}{2} \cdot \frac{1}{2^2} = 45/4 = 11.25 \)
- For \( r = 3 \): \( C_3 = \binom{10}{3} \cdot \frac{1}{2^3} = 120/8 = 15 \)

The maximum coefficient occurs at \( r = 3 \).

### Conclusion:
The power of \( x \) which has the greatest coefficient in the expansion of \( \left(1 + \frac{x}{2}\right)^{10} \) is \( x^3 \).