Question

The integer just greater than (3+√(5))²n is divisible by (n∈N)

• A. 2ⁿ-1
• B. 2ⁿ+1
• C. 2ⁿ+2
• D. Not divisible by 2

Hint:

Expressions involving a+√(b) and a-√(b) often give integers when added.

Answer 1

The Correct Option is A

Step 1: (3+√(5))²n + (3-√(5))²n is an integer. Step 2: Since 0<3-√(5)<1, 0<(3-√(5))²n<1 Step 3: Hence, the integer just greater than (3+√(5))²n is: (3+√(5))²n+(3-√(5))²n Step 4: This integer is divisible by 2ⁿ-1. Therefore, option (A) is correct.