The number of terms in the expansion of `(x+a)^(46)-(x-a)^(46)` after simplification is

Options

A. 23
B. 24
C. 46
D. 92

Answer

To find the number of terms in the expression \((x+a)^{46} - (x-a)^{46}\) after simplification, we can follow these steps:

### Step 1: Expand both expressions using the Binomial Theorem
Using the Binomial Theorem, we can expand both \((x+a)^{46}\) and \((x-a)^{46}\):

\[ (x+a)^{46} = \sum_{k=0}^{46} \binom{46}{k} x^{46-k} a^k \]

\[ (x-a)^{46} = \sum_{k=0}^{46} \binom{46}{k} x^{46-k} (-a)^k = \sum_{k=0}^{46} \binom{46}{k} x^{46-k} (-1)^k a^k \]

### Step 2: Write the difference of the two expansions
Now, we will find the difference:

\[ (x+a)^{46} - (x-a)^{46} = \sum_{k=0}^{46} \binom{46}{k} x^{46-k} a^k - \sum_{k=0}^{46} \binom{46}{k} x^{46-k} (-1)^k a^k \]

### Step 3: Combine the two expansions
Combining these two expansions gives:

\[ (x+a)^{46} - (x-a)^{46} = \sum_{k=0}^{46} \binom{46}{k} x^{46-k} (a^k - (-1)^k a^k) \]

### Step 4: Simplify the expression
Notice that \(a^k - (-1)^k a^k\) simplifies to:
- \(2a^k\) when \(k\) is odd (since \((-1)^k = -1\))
- \(0\) when \(k\) is even (since \((-1)^k = 1\))

Thus, the expression simplifies to:

\[ = \sum_{\text{odd } k} 2 \binom{46}{k} x^{46-k} a^k \]

### Step 5: Identify the number of terms
The terms that remain after simplification are those where \(k\) is odd. The odd values of \(k\) from \(0\) to \(46\) are \(1, 3, 5, \ldots, 45\).

To find how many odd numbers are there from \(1\) to \(46\):
- The sequence of odd numbers is an arithmetic sequence where the first term \(a = 1\) and the common difference \(d = 2\).
- The last term \(l = 45\).

The number of terms \(n\) in this sequence can be calculated using the formula for the \(n\)-th term of an arithmetic sequence:

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

Plugging in the values:

\[ 45 = 1 + (n-1) \cdot 2 \]

Solving for \(n\):

\[ 45 - 1 = (n-1) \cdot 2 \\ 44 = (n-1) \cdot 2 \\ n-1 = 22 \\ n = 23 \]

### Conclusion
Thus, the number of terms in the expansion of \((x+a)^{46} - (x-a)^{46}\) after simplification is \(23\).

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