# Quantitative Aptitude – Algebra – Functions – For any positive integer n, let f(n)

## For any positive integer n, let f(n) – Video

Q. 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?

Solution: Case 1) If m is odd, then m+1 will be even
So f(m) = m+3 and f(m+1) = m(m+1)
Given, 8f(m + 1) – f(m) = 2
Or 8* (m+2)(m+1)- (m+3) = 2
8m^2 + 24m + 16 – m – 3 =2
8m^2 + 23m – 11 = 0
No integer solution so m is not odd means m is even.
Case 2) if m is even , m+1 = odd
So f(m+1) = m+3 and f(m) = m(m+1)
Given, 8f(m + 1) – f(m) = 2
Or 8(m+1 + 3) – m(m+1) =2
8m + 32 – m^2 –m =2
m^2 -7m -30 =0
(m+3)*(m-10)=0
M = 10 or -3
As m is even so m =10

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