2002 AMC 10P Problems/Problem 5

Problem

Let $(a_n)_{n \geq 1}$ be a sequence such that $a_1 = 1$ and $3a_{n+1} - 3a_n = 1$ for all $n \geq 1.$ Find $a_{2002}.$

$\text{(A) }666 \qquad \text{(B) }667 \qquad \text{(C) }668 \qquad \text{(D) }669 \qquad \text{(E) }670$

Solution 1

The recursive rule is equal to $a_{n+1}=\frac{1}{3}+a_{n}$ for all $n \geq 1.$ By recursion, $a_{n+2}=\frac{1}{3}+a_{n+1}=a_{n+2}=\frac{1}{3}+\frac{1}{3}+a_n.$ If we set $n=1$ and repeat this process $2001$ times, we will get $a_{2001+1}=\frac{1}{3}(2001) + a_1=667+1=668.$

Thus, our answer is $\boxed{\textbf{(C) } 668}.$

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

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2002 AMC 10P Problems/Problem 5

Problem

Solution 1

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