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

Case 1: m is even.

Given, 8f(m + 1) – f(m) = 2

$\begin{array}{cc}& \Rightarrow 8(m+1+3)-m(m+1)=2\\ & \Rightarrow 8m+32-m^2-m=2\\ & \Rightarrow m^2-7m+30=0\\ & \Rightarrow (m-10)(m+3)=0\\ & \Rightarrow m=10or-3\end{array}$As m is positive integer, the only possible value of m =10.

Case 2:

If m is odd, then we would not be getting positive solution.