The sum of 1-1+1-1+1-1. . . To even number of terms is

Options

A. 2
B. zero
C. `-1`
D. `+1`

Explanation

To find the sum of the series \(1 - 1 + 1 - 1 + 1 - 1 + \ldots\) up to an even number of terms, we can follow these steps:

### Step 1: Understand the Pattern
The series alternates between 1 and -1. If we write out the first few terms, we see:
- 1st term: \(1\)
- 2nd term: \(1 - 1 = 0\)
- 3rd term: \(1 - 1 + 1 = 1\)
- 4th term: \(1 - 1 + 1 - 1 = 0\)
- 5th term: \(1 - 1 + 1 - 1 + 1 = 1\)
- 6th term: \(1 - 1 + 1 - 1 + 1 - 1 = 0\)

### Step 2: Analyze the Even Number of Terms
Since we are interested in an even number of terms, let’s denote the number of terms as \(2n\) (where \(n\) is a positive integer).

### Step 3: Calculate the Sum for Even Terms
For any even number of terms:
- The sum can be grouped as follows:
\[ (1 - 1) + (1 - 1) + \ldots + (1 - 1) \]
- Each pair \( (1 - 1) \) equals \(0\).

### Step 4: Count the Pairs
Since we have \(2n\) terms, we can form \(n\) pairs of \( (1 - 1) \):
- Therefore, the total sum is:
\[ n \times 0 = 0 \]

### Conclusion
Thus, the sum of the series \(1 - 1 + 1 - 1 + \ldots\) for an even number of terms is \(0\).

### Final Answer
The answer is \(0\).

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