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
Correct Answer: 10
Case 1: m is even.
Given, 8f(m + 1) - f(m) = 2
$\begin{align} & \Rightarrow 8\left( m+1+3 \right)-m\left( m+1 \right)=2 \\ & \Rightarrow 8m+32-{{m}^{2}}-m=2 \\ & \Rightarrow {{m}^{2}}-7m+30=0 \\ & \Rightarrow (m-10)(m+3)=0 \\ & \Rightarrow m=10\ or\ -3 \\ \end{align}$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.
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