For `|x| lt 1/2` , the value of the fourth term of `(1 - 2x)^(-3//4)` is

Options

A. `-77/16 x^3`
B. `16/77 x^3`
C. `77/16 x^3`
D. `-16/77x^3`

Answer

To find the fourth term of the expansion of \((1 - 2x)^{-3/4}\) for \(|x| < \frac{1}{2}\), we will use the Binomial Theorem for negative exponents. The Binomial Theorem states that:

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

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

### Step-by-Step Solution:

1. **Identify the parameters**:
We have \(n = -\frac{3}{4}\) and \(u = -2x\).

2. **Write the general term**:
The general term \(T_k\) in the expansion is given by:
\[ T_k = \binom{n}{k} u^k = \binom{-\frac{3}{4}}{k} (-2x)^k \]

3. **Calculate the fourth term**:
The fourth term corresponds to \(k = 3\) (since we start counting from \(k = 0\)):
\[ T_3 = \binom{-\frac{3}{4}}{3} (-2x)^3 \]

4. **Calculate the binomial coefficient**:
\[ \binom{-\frac{3}{4}}{3} = \frac{-\frac{3}{4} \left(-\frac{3}{4}-1\right) \left(-\frac{3}{4}-2\right)}{3!} \]
Simplifying this:
\[ = \frac{-\frac{3}{4} \left(-\frac{7}{4}\right) \left(-\frac{11}{4}\right)}{6} \]
\[ = \frac{3 \cdot 7 \cdot 11}{4 \cdot 4 \cdot 4 \cdot 6} \]
\[ = \frac{231}{384} \]

5. **Calculate \((-2x)^3\)**:
\[ (-2x)^3 = -8x^3 \]

6. **Combine the results**:
\[ T_3 = \frac{231}{384} \cdot (-8x^3) = -\frac{231 \cdot 8}{384} x^3 \]
\[ = -\frac{1848}{384} x^3 \]
Simplifying \(-\frac{1848}{384}\):
\[ = -\frac{231}{48} x^3 \]

### Final Answer:
Thus, the fourth term of the expansion of \((1 - 2x)^{-3/4}\) is:
\[ -\frac{231}{48} x^3 \]