Please submit your concern

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 8 f(m + 1) − f(m) = 2, then m equals

Explanation:

Case I: m is odd.

So, (m + 1) is even.  

∴ 8[(m + 1)(m + 2)] − (m + 3) = 2

∴ 8m² + 23m + 11 = 0.

Both roots of this equation are negative as sum of the roots (−23/8) is negative and the product (11/8) is positive. But it is given that m is a positive integer. Hence this case is discarded.

Case II: m is even.

So, (m + 1) is odd.

∴ 8(m + 3 + 1) − m(m + 1) = 2.

∴ m² − 7m − 30 = 0

Solving this equation, we get; m = 10 or −3.

Since m is positive, m = 10.  

Hence, 10.