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
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
ANSWER
Case 1: m is even.
Given, 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=10or−3⇒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 −3As 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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