The fourth term of the G.P. 4, - 2 , 1 , … is

Options

A. `(-1)/(2)`
B. `(1)/(2)`
C. `-1`
D. `(1)/(4)`

Answer

To find the fourth term of the geometric progression (G.P.) given by the terms 4, -2, 1, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the first term (a)**:
The first term of the G.P. is given as \( a = 4 \).

**Hint**: The first term is always the starting value of the sequence.

2. **Identify the second term and find the common ratio (r)**:
The second term is \( -2 \). The common ratio \( r \) can be found by dividing the second term by the first term:
\[ r = \frac{\text{second term}}{\text{first term}} = \frac{-2}{4} = -\frac{1}{2} \]

**Hint**: The common ratio can be found by dividing any term by the previous term.

3. **Use the formula for the nth term of a G.P.**:
The formula for the nth term of a G.P. is given by:
\[ T_n = a \cdot r^{n-1} \]
To find the fourth term (\( n = 4 \)):
\[ T_4 = a \cdot r^{4-1} = a \cdot r^3 \]

**Hint**: Remember that \( n \) is the position of the term you want to find.

4. **Substitute the values of \( a \) and \( r \)**:
Now substitute \( a = 4 \) and \( r = -\frac{1}{2} \):
\[ T_4 = 4 \cdot \left(-\frac{1}{2}\right)^3 \]

**Hint**: Make sure to calculate the power of the common ratio correctly.

5. **Calculate \( r^3 \)**:
Calculate \( \left(-\frac{1}{2}\right)^3 \):
\[ \left(-\frac{1}{2}\right)^3 = -\frac{1}{8} \]

**Hint**: When raising a negative number to an odd power, the result will be negative.

6. **Final calculation**:
Now substitute back into the equation for \( T_4 \):
\[ T_4 = 4 \cdot \left(-\frac{1}{8}\right) = -\frac{4}{8} = -\frac{1}{2} \]

**Hint**: Simplify the fraction to get the final answer.

### Conclusion:
The fourth term of the G.P. is \( -\frac{1}{2} \).