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

Solve the equation: 
\(8f(m + 1) - f(m) = 2\) 
where \(f(m) = m(m + 1)\) when \(m\) is even, and \(f(m) = m + 3\) when \(m\) is odd.

Case 1: When \(m\) is even

Since \(m\) is even, we use: \(f(m) = m(m + 1)\)
and \(m + 1\) is odd, so: \(f(m + 1) = (m + 1) + 3 = m + 4\)
 

Substitute into the original equation: 
 

\[8f(m + 1) - f(m) = 8(m + 4) - m(m + 1)\]

 

\[= 8m + 32 - (m^2 + m)\]

 

\[= -m^2 + 7m + 32\]

 Set equal to 2: 
 

\[-m^2 + 7m + 32 = 2 \Rightarrow m^2 - 7m + 30 = 0\]

Solve the quadratic: 

\[(m - 10)(m + 3) = 0 \Rightarrow m = 10 \text{ or } m = -3\]

 Only the positive integer solution is: \(\boxed{m = 10}\)

Case 2: When \(m\) is odd

Then \(f(m) = m + 3\) and \(f(m + 1) = (m + 1)(m + 2)\) since \(m + 1\) is even.

Substituting into the equation: 

\[8f(m + 1) - f(m) = 8(m + 1)(m + 2) - (m + 3)\]

Set equal to 2: 

\[8(m^2 + 3m + 2) - (m + 3) = 8m^2 + 24m + 16 - m - 3 = 8m^2 + 23m + 13\]

 Set equal to 2: 

\[8m^2 + 23m + 13 = 2 \Rightarrow 8m^2 + 23m + 11 = 0\]

 The discriminant is: 

\[D = 23^2 - 4 \cdot 8 \cdot 11 = 529 - 352 = 177\]

 Since the discriminant is not a perfect square, there is no integer solution.

Conclusion:

The only valid solution is when \(m = 10\). 
Final Answer: \(\boxed{m = 10}\)