Find the 7th term of the series `-1/8 +1/4 - 1/2 + 1`… :

Options

A. 8
B. `-16`
C. `-8`
D. none of these

Answer

To find the 7th term of the series \(-\frac{1}{8} + \frac{1}{4} - \frac{1}{2} + 1 + \ldots\), we can follow these steps:

### Step 1: Identify the first term and the common ratio
The first term \(a\) of the series is:
\[ a = -\frac{1}{8} \]

Next, we need to find the common ratio \(r\). We can find \(r\) by dividing the second term by the first term:
\[ r = \frac{\frac{1}{4}}{-\frac{1}{8}} = \frac{1}{4} \times \left(-\frac{8}{1}\right) = -2 \]

### Step 2: Verify the common ratio with another pair of terms
To ensure that the common ratio is consistent, we can check the ratio of the third term to the second term:
\[ r = \frac{-\frac{1}{2}}{\frac{1}{4}} = -\frac{1}{2} \times 4 = -2 \]
This confirms that the common ratio \(r = -2\).

### Step 3: Use the formula for the nth term of a geometric progression
The formula for the nth term \(T_n\) of a geometric progression is given by:
\[ T_n = a \cdot r^{n-1} \]
For the 7th term, we set \(n = 7\):
\[ T_7 = a \cdot r^{7-1} = -\frac{1}{8} \cdot (-2)^{6} \]

### Step 4: Calculate \((-2)^6\)
Calculating \((-2)^6\):
\[ (-2)^6 = 64 \]

### Step 5: Substitute back into the formula
Now, substituting back into the formula for \(T_7\):
\[ T_7 = -\frac{1}{8} \cdot 64 \]

### Step 6: Simplify the expression
Now, simplifying:
\[ T_7 = -\frac{64}{8} = -8 \]

### Conclusion
The 7th term of the series is:
\[ \boxed{-8} \]