The middle term in the expansion of `(x/2+2)^8` is 1120, then `x in R` is equal to a. -2 b. 3 c. `-3` d. `2`

Options

A. `-2`
B. `3`
C. `-3`
D. `2`

Explanation

To find the value of \( x \) in the expression \( \left(\frac{x}{2} + 2\right)^8 \) such that the middle term is 1120, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the total number of terms**:
The total number of terms in the expansion of \( (a + b)^n \) is \( n + 1 \). Here, \( n = 8 \), so the total number of terms is \( 8 + 1 = 9 \).

**Hint**: Remember that the number of terms in a binomial expansion is always \( n + 1 \).

2. **Determine the middle term**:
Since the total number of terms is odd (9), the middle term is the \( \frac{n + 1}{2} \)th term, which is the 5th term.

**Hint**: For an odd number of terms, the middle term can be found using \( \frac{n + 1}{2} \).

3. **Write the general term**:
The general term \( T_{r+1} \) in the expansion of \( (a + b)^n \) is given by:
\[ T_{r+1} = \binom{n}{r} a^{n-r} b^r \]
Here, \( a = \frac{x}{2} \), \( b = 2 \), and \( n = 8 \). For the 5th term, \( r = 4 \):
\[ T_5 = \binom{8}{4} \left(\frac{x}{2}\right)^{8-4} \cdot 2^4 \]

**Hint**: Use the binomial coefficient formula \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \).

4. **Calculate the 5th term**:
Substitute the values:
\[ T_5 = \binom{8}{4} \left(\frac{x}{2}\right)^4 \cdot 2^4 \]
\[ = \binom{8}{4} \cdot \frac{x^4}{16} \cdot 16 \]
\[ = \binom{8}{4} \cdot x^4 \]

5. **Calculate \( \binom{8}{4} \)**:
\[ \binom{8}{4} = \frac{8!}{4!4!} = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = 70 \]
Therefore, the 5th term becomes:
\[ T_5 = 70 x^4 \]

**Hint**: Factorials can be simplified by cancelling common terms.

6. **Set the equation**:
We know that this term equals 1120:
\[ 70 x^4 = 1120 \]

7. **Solve for \( x^4 \)**:
Divide both sides by 70:
\[ x^4 = \frac{1120}{70} = 16 \]

**Hint**: Always simplify fractions to make calculations easier.

8. **Find \( x \)**:
Taking the fourth root of both sides gives:
\[ x = \pm 2 \]

**Hint**: Remember that taking an even root results in both positive and negative solutions.

9. **Check the options**:
The possible values of \( x \) are \( 2 \) and \( -2 \). The options given were:
a. -2
b. 3
c. -3
d. 2

Hence, the correct answers are \( x = -2 \) or \( x = 2 \).

### Final Answer:
The values of \( x \) in \( \mathbb{R} \) are \( -2 \) and \( 2 \).