The number of terms in the expansion of `(x+a)^(53)+(x-a)^(53)` after simplification is

Options

A. 106
B. 26
C. 27
D. 53

Answer

To find the number of terms in the expansion of \((x + a)^{53} + (x - a)^{53}\) after simplification, we will follow these steps:

### Step 1: Expand both expressions using the Binomial Theorem
The Binomial Theorem states that:
\[ (x + a)^n = \sum_{r=0}^{n} \binom{n}{r} x^{n-r} a^r \]
For \((x + a)^{53}\):
\[ (x + a)^{53} = \sum_{r=0}^{53} \binom{53}{r} x^{53 - r} a^r \]

For \((x - a)^{53}\):
\[ (x - a)^{53} = \sum_{r=0}^{53} \binom{53}{r} x^{53 - r} (-a)^r = \sum_{r=0}^{53} \binom{53}{r} x^{53 - r} (-1)^r a^r \]

### Step 2: Add the two expansions
Now we add the two expansions:
\[ (x + a)^{53} + (x - a)^{53} = \sum_{r=0}^{53} \binom{53}{r} x^{53 - r} a^r + \sum_{r=0}^{53} \binom{53}{r} x^{53 - r} (-1)^r a^r \]

### Step 3: Simplify the expression
When we combine the two sums, we notice that terms where \(r\) is odd will cancel out, while terms where \(r\) is even will double:
\[ = \sum_{r \text{ even}} \binom{53}{r} x^{53 - r} (a^r + (-1)^r a^r) = \sum_{r \text{ even}} 2 \binom{53}{r} x^{53 - r} a^r \]

### Step 4: Identify the even values of \(r\)
The even values of \(r\) can be \(0, 2, 4, \ldots, 52\). The maximum value of \(r\) is \(53\), which is odd, so the highest even \(r\) is \(52\).

### Step 5: Count the number of even terms
The even integers from \(0\) to \(52\) can be expressed as:
\[ 0, 2, 4, 6, \ldots, 52 \]
This is an arithmetic progression where:
- First term \(a = 0\)
- Last term \(l = 52\)
- Common difference \(d = 2\)

To find the number of terms \(n\) in this sequence, we can use the formula for the \(n\)-th term of an arithmetic progression:
\[ l = a + (n-1)d \]
Substituting the known values:
\[ 52 = 0 + (n-1) \cdot 2 \]
\[ 52 = (n-1) \cdot 2 \]
\[ n - 1 = 26 \]
\[ n = 27 \]

### Final Answer
Thus, the number of terms in the expansion of \((x + a)^{53} + (x - a)^{53}\) after simplification is **27**.

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