Question

The number of terms in the expansion of \( (1 + 5\sqrt{2}x)^9 + (1 - 5\sqrt{2}x)^9 \) is:

• A. 5
• B. 7
• C. 9
• D. 10

Hint:

When adding binomial expansions, terms with opposite signs cancel out. Only terms with the same powers of \( x \) that have the same sign will remain.

Answer 1

The Correct Option is A

We are given the expression: \[ (1 + 5\sqrt{2}x)^9 + (1 - 5\sqrt{2}x)^9 \] To find the number of terms in the expansion, we use the binomial theorem. The binomial expansion of \( (1 + 5\sqrt{2}x)^9 \) and \( (1 - 5\sqrt{2}x)^9 \) will contain terms of the form: \[ \binom{9}{k} (5\sqrt{2}x)^k \] Expanding each expression: \[ (1 + 5\sqrt{2}x)^9 = \sum_{k=0}^{9} \binom{9}{k} (5\sqrt{2}x)^k \] \[ (1 - 5\sqrt{2}x)^9 = \sum_{k=0}^{9} \binom{9}{k} (-5\sqrt{2}x)^k \] When adding the two expansions, terms where \( k \) is odd will cancel out, because the powers of \( x \) will have opposite signs (due to the \( -5\sqrt{2}x \) term), and terms where \( k \) is even will add up. Thus, only the even terms in both expansions will remain. The even values of \( k \) are \( k = 0, 2, 4, 6, 8 \), so there are 5 terms in the expansion. Thus, the correct number of terms is 5. The correct answer is Option A.