Question

The total number of terms in the expansion of \((x + a)^{100} + (x - a)^{100}\) after simplification will be:

• A. \(202\)
• B. \(51\)
• C. \(50\)
• D. None of these

Hint:

In expressions like ((x+a)^n + (x-a)^n):

Odd-power terms cancel out
Only even-power terms remain
Number of terms (= fracn2 + 1) when (n) is even

Answer 1

The Correct Option is B

Step 1: Write the general expansions. Using the binomial theorem: \[ (x+a)^{100} = \sum_{k=0}^{100} \binom{100}{k} x^{100-k} a^{k} \] \[ (x-a)^{100} = \sum_{k=0}^{100} \binom{100}{k} x^{100-k} (-a)^{k} \] Step 2: Add the two expansions. \[ (x+a)^{100} + (x-a)^{100} = \sum_{k=0}^{100} \binom{100}{k} x^{100-k} \left[a^{k} + (-a)^{k}\right] \] Step 3: Analyze the parity of powers of \(a\).

If \(k\) is odd: \(a^{k} + (-a)^{k} = 0\)
If \(k\) is even: \(a^{k} + (-a)^{k} = 2a^{k}\)
Hence, only terms with even powers of \(a\) survive. Step 4: Count the surviving terms. Even values of \(k\) from \(0\) to \(100\) are: \[ 0, 2, 4, \ldots, 100 \] Number of even integers from \(0\) to \(100\): \[ = \frac{100}{2} + 1 = 50 + 1 = 51 \] Step 5: Final conclusion. After simplification, the expansion contains \(\boxed{51}\) distinct terms.