Question

For any positive integer n, let f(n) = n(n + 1) if n is even, and f(n) = n + 3 if n is odd. If m is a positive integer such that 8f(m + 1) - f(m) = 2, then m equals

• A. 13
• B. 12
• C. 11
• D. 10

Answer 1

The Correct Option is D

For Case 1, when m is even,
the equation is given as 8f(m + 1) - f(m) = 2.
⇒ 8( m + 1 + 3) - m(m + 1) = 2
⇒ 8m + 32 - m² - m = 2
⇒ m² - 7m + 30 = 0
⇒ (m - 10)(m + 3) = 0
⇒ m = 10 or -3
Since m must be a positive integer, the only valid solution is m = 10.
In Case 2, when m is odd, there is no positive solution for the given equation.