Write down the first three terms is the following expansions
`(1 + x/2)^(-5)`

Answer

To find the first three terms of the expansion of \((1 + \frac{x}{2})^{-5}\), we can use the Binomial Theorem for negative exponents. The Binomial Theorem states that:

\[ (1 + x)^n = \sum_{r=0}^{\infty} \binom{n}{r} x^r \]

where \(\binom{n}{r} = \frac{n(n-1)(n-2)...(n-r+1)}{r!}\).

In our case, we have \(n = -5\) and \(x = \frac{x}{2}\). We will calculate the first three terms of the expansion.

### Step 1: Identify the first three terms in the expansion

1. **First Term**: For \(r = 0\):
\[ \binom{-5}{0} \left(\frac{x}{2}\right)^0 = 1 \]

2. **Second Term**: For \(r = 1\):
\[ \binom{-5}{1} \left(\frac{x}{2}\right)^1 = -5 \cdot \frac{x}{2} = -\frac{5x}{2} \]

3. **Third Term**: For \(r = 2\):
\[ \binom{-5}{2} \left(\frac{x}{2}\right)^2 = \frac{-5 \cdot (-6)}{2} \cdot \left(\frac{x}{2}\right)^2 = \frac{30}{2} \cdot \frac{x^2}{4} = \frac{15}{4} x^2 \]

### Step 2: Combine the terms

Now, we combine these terms to write down the first three terms of the expansion:

\[ 1 - \frac{5x}{2} + \frac{15}{4} x^2 \]

### Final Result

Thus, the first three terms of the expansion of \((1 + \frac{x}{2})^{-5}\) are:

\[ 1, -\frac{5x}{2}, \frac{15}{4} x^2 \]

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