The number of terms in the expansion of `( x +a)^(100) + ( x -a)^(100)` after simplification is `:`

Options

A. 202
B. 101
C. 51
D. 50

Answer

To find the number of terms in the expansion of \( (x + a)^{100} + (x - a)^{100} \) after simplification, we can follow these steps:

### Step 1: Understand the Binomial Expansion
The binomial expansion of \( (x + a)^n \) is given by:
\[ (x + a)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} a^k \]
Similarly, for \( (x - a)^n \):
\[ (x - a)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} (-a)^k \]

### Step 2: Combine the Two Expansions
Now, we need to combine the expansions:
\[ (x + a)^{100} + (x - a)^{100} = \sum_{k=0}^{100} \binom{100}{k} x^{100-k} a^k + \sum_{k=0}^{100} \binom{100}{k} x^{100-k} (-a)^k \]

### Step 3: Simplify the Combined Expression
When we add these two expansions, terms with odd powers of \( a \) will cancel out because \( a^k \) and \( (-a)^k \) will have opposite signs for odd \( k \). Therefore, only the even powers of \( a \) will remain.

### Step 4: Identify the Remaining Terms
The remaining terms correspond to even values of \( k \). The even values of \( k \) from \( 0 \) to \( 100 \) are:
\[ 0, 2, 4, \ldots, 100 \]
This forms an arithmetic sequence where the first term \( a = 0 \), the last term \( l = 100 \), and the common difference \( d = 2 \).

### Step 5: Calculate the Number of Terms
To find the number of terms in this sequence, we can use the formula for the number of terms in an arithmetic sequence:
\[ n = \frac{l - a}{d} + 1 \]
Substituting the values:
\[ n = \frac{100 - 0}{2} + 1 = 50 + 1 = 51 \]

### Conclusion
Thus, the number of terms in the expansion of \( (x + a)^{100} + (x - a)^{100} \) after simplification is \( 51 \).

### Final Answer
The number of terms is \( \boxed{51} \).

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