Question

Consider the sequence of positive integers

$\begin{gathered} 1, 2, 1, 2, 3, 2, 1, 2, 3, 4, \\ 3, 2, 1, 2, 3, 4, 5, 4, 3, 2, \\ 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, \\ 1, 2, \ldots \end{gathered}$

What is the $2025$ th term in this sequence?

$5$

$15$

$16$

$44$

$45$

Answer: E

Concepts:pattern recognition perfect square

Difficulty rating: 1200

Solution:

Group the sequence into blocks. Block $k$ reads $k, k-1, \ldots, 2, 1, 2, \ldots, k-1, k,$ which is $2k - 1$ terms and ends on $k.$ So after block $k$ we've used $1 + 3 + \cdots + (2k-1) = k^2$ terms. Notice $2025 = 45^2.$ That's exactly the end of block $45,$ whose last term is $45.$ Thus, E is the correct answer.

2025 AMC 10A Problem 5

Answer: E

Difficulty rating: 1200

Solution:

Problem 5 in Other Years