Question

The integer just greater than \((3 + \sqrt{5})^{2n}\) is divisible by

• A. \(2^{\,n-1}\)
• B. \(2^{\,n+1}\)
• C. \(2^{\,n+2}\)
• D. Not divisible by 2

Hint:

Use conjugates to evaluate expressions with irrational powers.

Answer 1

The Correct Option is B

Step 1: Expand using the binomial theorem:
\[ (3 + \sqrt{5})^{2n} + (3 - \sqrt{5})^{2n} \] is an integer.
Step 2: Since \(0 < (3 - \sqrt{5})^{2n} < 1\),
\[ \lceil (3 + \sqrt{5})^{2n} \rceil = (3 + \sqrt{5})^{2n} + (3 - \sqrt{5})^{2n} \]
Step 3: The sum is divisible by \(2^n + 1\).