Difference between revisions of "2017 AMC 12A Problems/Problem 7"
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− | Define a function on the positive integers recursively by <math>f(1) = 2</math>, <math>f(n) = f(n-1) + | + | ==Problem== |
+ | |||
+ | Define a function on the positive integers recursively by <math>f(1) = 2</math>, <math>f(n) = f(n-1) + 1</math> if <math>n</math> is even, and <math>f(n) = f(n-2) + 2</math> if <math>n</math> is odd and greater than <math>1</math>. What is <math>f(2017)</math>? | ||
+ | |||
+ | <math> \textbf{(A)}\ 2017 \qquad\textbf{(B)}\ 2018 \qquad\textbf{(C)}\ 4034 \qquad\textbf{(D)}\ 4035 \qquad\textbf{(E)}\ 4036 </math> | ||
+ | ==Solution== | ||
+ | This is a recursive function, which means the function is used to evaluate itself. To solve this, we must identify the base case, <math>f(1)=2</math>. We also know that when <math>n</math> is odd, <math>f(n)=f(n-2)+2</math>. Thus we know that <math>f(2017)=f(2015)+2</math>. Thus we know that n will always be odd in the recursion of <math>f(2017)</math>, and we add <math>2</math> each recursive cycle, which there are <math>1008</math> of. Thus the answer is <math>1008*2+2=2018</math>, which is answer | ||
+ | <math>\boxed{\textbf{(B)}}</math>. | ||
+ | Note that when you write out a few numbers, you find that <math>f(n)=n+1</math> for any <math>n</math>, so <math>f(2017)=2018</math> | ||
+ | |||
+ | ==See Also== | ||
+ | {{AMC12 box|year=2017|ab=A|num-b=6|num-a=8}} |
Latest revision as of 10:02, 16 April 2019
Problem
Define a function on the positive integers recursively by , if is even, and if is odd and greater than . What is ?
Solution
This is a recursive function, which means the function is used to evaluate itself. To solve this, we must identify the base case, . We also know that when is odd, . Thus we know that . Thus we know that n will always be odd in the recursion of , and we add each recursive cycle, which there are of. Thus the answer is , which is answer . Note that when you write out a few numbers, you find that for any , so
See Also
2017 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 6 |
Followed by Problem 8 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |