Difference between revisions of "2002 AMC 10P Problems/Problem 5"
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| − | == Solution 1== | + | == Solution 1 == |
The recursive rule is equal to <math>a_{n+1}=\frac{1}{3}+a_{n}</math> for all <math>n \geq 1.</math> By recursion, <math>a_{n+2}=\frac{1}{3}+a_{n+1}=a_{n+2}=\frac{1}{3}+\frac{1}{3}+a_n.</math> If we set <math>n=1</math> and repeat this process <math>2001</math> times, we will get <math>a_{2001+1}=a_{2002}=\frac{1}{3}(2001) + a_1=667+1=668.</math> | The recursive rule is equal to <math>a_{n+1}=\frac{1}{3}+a_{n}</math> for all <math>n \geq 1.</math> By recursion, <math>a_{n+2}=\frac{1}{3}+a_{n+1}=a_{n+2}=\frac{1}{3}+\frac{1}{3}+a_n.</math> If we set <math>n=1</math> and repeat this process <math>2001</math> times, we will get <math>a_{2001+1}=a_{2002}=\frac{1}{3}(2001) + a_1=667+1=668.</math> | ||
| + | |||
| + | Thus, our answer is <math>\boxed{\textbf{(C) } 668}.</math> | ||
| + | |||
| + | == Solution 2 == | ||
| + | Find the first few terms of the sequence and find a pattern. | ||
| + | <cmath> | ||
| + | \begin{tabular}{|c|c|c|c|c|c|} | ||
| + | \hline | ||
| + | n & 1 & 2 & 3 & 4 & 5 \\ | ||
| + | \hline | ||
| + | an & 1 & \frac{4}{3} & \frac{5/3} & 2 & \frac{7/3} \\ | ||
| + | \hline | ||
| + | \end{tabular} | ||
| + | </cmath> | ||
| + | |||
| + | From here, we can deduce that <math>a_n=\frac{2}{3}+\frac{1}{3}n.</math> Plugging in <math>n=2002,</math> <math>a_2002=\frac{2}{3}+\frac{2002}{3}=668.</math> | ||
Thus, our answer is <math>\boxed{\textbf{(C) } 668}.</math> | Thus, our answer is <math>\boxed{\textbf{(C) } 668}.</math> | ||
Revision as of 19:54, 14 July 2024
Contents
Problem
Let 
be a sequence such that 
and 
for all 
Find 
Solution 1
The recursive rule is equal to 
for all By recursion, 
If we set 
and repeat this process 
times, we will get 
Thus, our answer is 
Solution 2
Find the first few terms of the sequence and find a pattern.
\[\begin{tabular}{|c|c|c|c|c|c|}
\hline
n & 1 & 2 & 3 & 4 & 5 \\
\hline
an & 1 & \frac{4}{3} & \frac{5/3} & 2 & \frac{7/3} \\
\hline
\end{tabular}\] (Error compiling LaTeX. Unknown error_msg)
From here, we can deduce that 
Plugging in 
Thus, our answer is
See also
| 2002 AMC 10P (Problems • Answer Key • Resources) | ||
| Preceded by Problem 4 |
Followed by Problem 6 | |
| 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 10 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
