If the middle term in the expansion of `(p/2+2)^(8)` is `1120`, find `p`.

Options

A. `pm1`
B. `pm2`
C. `pm3`
D. `pm4`

Explanation

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

### Step 1: Identify the middle term
In the expansion of \( (a + b)^n \), the middle term is given by the formula:
\[ T_{r+1} = \binom{n}{r} a^{n-r} b^r \]
For \( n = 8 \), the middle term occurs at \( r = \frac{n}{2} = 4 \). Thus, the middle term is:
\[ T_{4+1} = T_5 = \binom{8}{4} \left(\frac{p}{2}\right)^{8-4} \cdot 2^4 \]

### Step 2: Calculate the binomial coefficient
Calculate \( \binom{8}{4} \):
\[ \binom{8}{4} = \frac{8!}{4! \cdot 4!} = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = 70 \]

### Step 3: Substitute values into the middle term formula
Now substitute \( \binom{8}{4} \), \( \left(\frac{p}{2}\right)^4 \), and \( 2^4 \) into the middle term:
\[ T_5 = 70 \left(\frac{p}{2}\right)^4 \cdot 2^4 \]
Since \( 2^4 = 16 \), we have:
\[ T_5 = 70 \left(\frac{p}{2}\right)^4 \cdot 16 \]

### Step 4: Simplify the expression
Now simplify the expression:
\[ T_5 = 70 \cdot 16 \cdot \left(\frac{p^4}{16}\right) = 70p^4 \]

### Step 5: Set the equation equal to 1120
We know that this middle term equals \( 1120 \):
\[ 70p^4 = 1120 \]

### Step 6: Solve for \( p^4 \)
Now divide both sides by \( 70 \):
\[ p^4 = \frac{1120}{70} = 16 \]

### Step 7: Solve for \( p \)
Taking the fourth root of both sides gives:
\[ p = \pm 2 \]

### Final Answer
Thus, the values of \( p \) are:
\[ p = 2 \quad \text{or} \quad p = -2 \]