Question

Number of terms in the binomial expansion of (x + a)⁵³ + (x - a)⁵³ is:

• A. 25
• B. 26
• C. 27
• D. 26.5

Hint:

In binomial expansions of sums and differences, terms with odd powers of the variable cancel out when added together.

Answer 1

The Correct Option is C

The expression is \( (x + a)^{53} + (x - a)^{53} \). The binomial expansion of \( (x + a)^{53} \) will have 54 terms (from \( x^{53} \) to \( a^{53} \)), and similarly for \( (x - a)^{53} \). Since the terms in \( (x + a)^{53} \) and \( (x - a)^{53} \) with odd powers of \( x \) will cancel out, we are left with only the even powers of \( x \). Step 1: Count the number of terms The powers of \( x \) in the expansion will be \( 0, 2, 4, \dots, 52 \). These are the even powers from 0 to 52, inclusive. The number of terms is: \[ \frac{52 - 0}{2} + 1 = 27 \] Thus, the correct answer is 27.