The total number of terms in the expansion of `(x+a)^(51) - ( x-a)^(51) ` after simplification is

Options

A. 102
B. 25
C. 26
D. None of these

Answer

To find the total number of terms in the expansion of \((x+a)^{51} - (x-a)^{51}\) after simplification, we can follow these steps:

### Step 1: Expand both expressions using the Binomial Theorem
Using the Binomial Theorem, we can expand \((x+a)^{51}\) and \((x-a)^{51}\):

\[ (x+a)^{51} = \sum_{k=0}^{51} \binom{51}{k} x^{51-k} a^k \]

\[ (x-a)^{51} = \sum_{k=0}^{51} \binom{51}{k} x^{51-k} (-a)^k = \sum_{k=0}^{51} \binom{51}{k} x^{51-k} (-1)^k a^k \]

### Step 2: Combine the two expansions
Now, we subtract the second expansion from the first:

\[ (x+a)^{51} - (x-a)^{51} = \sum_{k=0}^{51} \binom{51}{k} x^{51-k} a^k - \sum_{k=0}^{51} \binom{51}{k} x^{51-k} (-1)^k a^k \]

This simplifies to:

\[ = \sum_{k=0}^{51} \binom{51}{k} x^{51-k} a^k (1 - (-1)^k) \]

### Step 3: Analyze the terms
The expression \(1 - (-1)^k\) evaluates to:
- \(0\) when \(k\) is even (because \(1 - 1 = 0\))
- \(2\) when \(k\) is odd (because \(1 - (-1) = 2\))

Thus, only the terms where \(k\) is odd will remain in the expansion.

### Step 4: Identify the odd values of \(k\)
The odd values of \(k\) from \(0\) to \(51\) are \(1, 3, 5, \ldots, 51\). This forms an arithmetic progression (AP) where:
- The first term \(a = 1\)
- The common difference \(d = 2\)
- The last term \(l = 51\)

### Step 5: Calculate the number of odd terms
To find the number of terms in this AP, we can use the formula for the \(n\)-th term of an AP:

\[ l = a + (n-1) d \]

Substituting the known values:

\[ 51 = 1 + (n-1) \cdot 2 \]

Solving for \(n\):

\[ 51 - 1 = (n-1) \cdot 2 \]
\[ 50 = (n-1) \cdot 2 \]
\[ n-1 = 25 \]
\[ n = 26 \]

### Conclusion
Thus, the total number of terms in the expansion of \((x+a)^{51} - (x-a)^{51}\) after simplification is **26**.

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