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A function f is defined on the positive integers by f(1) = 1, and for each integer n greater than 1, f(n) = f(n−1) + n if n is even, and f(n) = f(n−1) − n if n is odd. What is f(50)?
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Correct answer
B
Computing all 50 steps directly would work, but it is faster to compute a handful of terms, spot the pattern at the even arguments, and then jump straight to n = 50.
Walk out the first few values: f(1) = 1; f(2) = 1 + 2 = 3; f(3) = 3 − 3 = 0; f(4) = 0 + 4 = 4; f(5) = 4 − 5 = −1; f(6) = −1 + 6 = 5; f(7) = 5 − 7 = −2; f(8) = −2 + 8 = 6.
Look only at the even arguments: f(2) = 3, f(4) = 4, f(6) = 5, f(8) = 6. Each step from one even argument to the next adds exactly 1. This happens because moving from f(2k) to f(2k + 2) means first subtracting the odd term (2k + 1) and then adding the next even term (2k + 2), and −(2k + 1) + (2k + 2) = 1. So f(2k) = k + 2.
For n = 50, k = 25, so f(50) = 25 + 2 = 27.
You can also confirm this by summing the signed terms directly: f(50) = 1 + (2 − 3 + 4 − 5 + 6 − 7 + ... + 48 − 49 + 50). Group the terms from 2 to 49 into consecutive pairs (2 − 3), (4 − 5), ..., (48 − 49); each pair equals −1, and there are 24 such pairs, with 50 left unpaired. That gives 24(−1) + 50 = 26, and f(50) = 1 + 26 = 27, which matches.
The answer is 27, choice B.
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