Which term of the sequence `1/2,-1/4,1/8,-1/16 …..is -1/256`?

Options

A. 9th
B. 8th
C. 7th
D. 5th

Answer

To find which term of the sequence \( \frac{1}{2}, -\frac{1}{4}, \frac{1}{8}, -\frac{1}{16}, \ldots \) is equal to \( -\frac{1}{256} \), we can follow these steps:

### Step 1: Identify the pattern in the sequence
The given sequence alternates between positive and negative terms. The numerators are all 1, while the denominators are powers of 2.

### Step 2: Write the general term of the sequence
The \( n \)-th term of the sequence can be expressed as:
\[ a_n = \frac{(-1)^{n+1}}{2^n} \]
Here, \( (-1)^{n+1} \) accounts for the alternating signs, and \( 2^n \) represents the denominators.

### Step 3: Set the general term equal to \( -\frac{1}{256} \)
We need to find \( n \) such that:
\[ \frac{(-1)^{n+1}}{2^n} = -\frac{1}{256} \]

### Step 4: Solve for \( n \)
1. **Equate the denominators**:
Since \( -\frac{1}{256} \) has a denominator of 256, we set:
\[ 2^n = 256 \]
We know that \( 256 = 2^8 \), so:
\[ n = 8 \]

2. **Check the sign**:
Now, we need to check if \( (-1)^{n+1} \) gives a negative sign when \( n = 8 \):
\[ (-1)^{8+1} = (-1)^9 = -1 \]
This confirms that the term is indeed negative.

### Conclusion
Thus, the term \( -\frac{1}{256} \) corresponds to the 8th term of the sequence.

### Final Answer
The answer is the 8th term.

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