Define a function on the positive integers recursively by f(1)=2, f(n)=f(n-1)+1 if n is even, and f(n)=f(n-2)+2 if n is odd and greater than 1. What is f(2017)?

2017 AMC 12A Problem 7

Attempt Problem 7 of the 2017 AMC 12A below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2017 AMC 12A solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

Define a function on the positive integers recursively by

f(1)=2,

f(n)=f(n-1)+1

n

is even, and

f(n)=f(n-2)+2

n

is odd and greater than

1.

What is

f(2017)?

2017

2018

4034

4035

4036

Compute

f(1),f(2),f(3),\ldots

and look for a pattern

Check that

f(n)=n+1

satisfies both recursive rules

Listing values:

f(1)=2,

f(2)=f(1)+1=3,

f(3)=f(1)+2=4,

f(4)=f(3)+1=5,

suggesting

f(n)=n+1.

Both rules are consistent with

f(n)=n+1:

for even

n,

(n-1)+1+1=n+1,

and for odd

n,

(n-2)+1+2=n+1.

Since the recursion determines

f

uniquely,

f(2017)=2018.

Thus, the correct answer is B.