The total number of terms in the expansion of `(x + a)^(200) + (x - a)^(200)` after simplification is

Options

A. 101
B. 102
C. 201
D. 202

Answer

To find the total number of terms in the expansion of \((x + a)^{200} + (x - a)^{200}\) after simplification, we can follow these steps:

### Step-by-Step Solution:

1. **Expand \((x + a)^{200}\)**:
The expansion of \((x + a)^{200}\) can be expressed using the Binomial Theorem:
\[ (x + a)^{200} = \sum_{r=0}^{200} \binom{200}{r} x^{200 - r} a^r \]

2. **Expand \((x - a)^{200}\)**:
Similarly, the expansion of \((x - a)^{200}\) is:
\[ (x - a)^{200} = \sum_{r=0}^{200} \binom{200}{r} x^{200 - r} (-a)^r = \sum_{r=0}^{200} \binom{200}{r} x^{200 - r} (-1)^r a^r \]

3. **Combine the Two Expansions**:
Now, we add the two expansions:
\[ (x + a)^{200} + (x - a)^{200} = \sum_{r=0}^{200} \binom{200}{r} x^{200 - r} a^r + \sum_{r=0}^{200} \binom{200}{r} x^{200 - r} (-1)^r a^r \]
This can be simplified to:
\[ = \sum_{r=0}^{200} \binom{200}{r} x^{200 - r} (1 + (-1)^r) a^r \]

4. **Identify Which Terms Survive**:
The term \(1 + (-1)^r\) will be:
- \(2\) if \(r\) is even (since \(1 + 1 = 2\))
- \(0\) if \(r\) is odd (since \(1 - 1 = 0\))

Therefore, only the even terms will survive in the combined expansion.

5. **Determine the Values of \(r\)**:
The even values of \(r\) range from \(0\) to \(200\). The even integers in this range are:
\[ 0, 2, 4, \ldots, 200 \]
This forms an arithmetic sequence where:
- First term \(a = 0\)
- Common difference \(d = 2\)
- Last term \(l = 200\)

6. **Count the Number of Even Terms**:
To find the number of terms in this sequence, we can use the formula for the \(n\)-th term of an arithmetic sequence:
\[ n = \frac{l - a}{d} + 1 = \frac{200 - 0}{2} + 1 = 100 + 1 = 101 \]

### Final Answer:
Thus, the total number of terms in the expansion of \((x + a)^{200} + (x - a)^{200}\) after simplification is **101**.