AMC 10 · 2025 · #5
Easy mode Grade 5Here is a list of numbers that climbs up and then walks back down, over and over:
1,2,1,2,3,2,1,2,3,4,3,2,1,2,3,4,5,4,3,2,1,2,…
Each "hill" counts up one step higher than the hill before, then counts back down to 1. What number sits in the 2025th spot?
Pick an answer.
AMC 10 2025 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: A sequence of positive integers climbs and falls in repeating "mountains": $1,2,1,2,3,2,1,2,3,4,3,2,1,\dots$. Find the value of the $2025$th term.
Givens: The sequence starts $1, 2, 1, 2, 3, 2, 1, 2, 3, 4, 3, 2, 1, 2, 3, 4, 5, 4, 3, 2, 1, \dots$; It rises $1,2,\dots,k$ then falls back to $1$, and each mountain reaches one higher than the last; Answer choices: (A) $5$, (B) $15$, (C) $16$, (D) $44$, (E) $45$
Unknowns: The value sitting at position $2025$
Understand
Restated: A sequence of positive integers climbs and falls in repeating "mountains": $1,2,1,2,3,2,1,2,3,4,3,2,1,\dots$. Find the value of the $2025$th term.
Givens: The sequence starts $1, 2, 1, 2, 3, 2, 1, 2, 3, 4, 3, 2, 1, 2, 3, 4, 5, 4, 3, 2, 1, \dots$; It rises $1,2,\dots,k$ then falls back to $1$, and each mountain reaches one higher than the last; Answer choices: (A) $5$, (B) $15$, (C) $16$, (D) $44$, (E) $45$
Plan
Primary tool: #5 Look for a Pattern
Secondary: #6 Guess and Check, #3 Eliminate Possibilities
Listing all 2025 terms is hopeless, so the win comes from finding a rule that jumps straight to any position. Tool #5 (Look for a Pattern) is built for repeating numbers like this. The key is to not track every term — instead watch only the positions that are perfect squares ($1, 4, 9, 16, 25, \dots$) and read off what value lands there. Once a clean rule appears, Tool #6 (Guess and Check) tests whether $2025$ is itself a perfect square, and Tool #3 (Eliminate Possibilities) guards against the classic off-by-one trap between the choices $44$ and $45$.
Execute — Answer: E
4.OA.C.5 Step 1 See the mountain structure
- Read the sequence in blocks.
- It climbs $1, 2, 3, \dots$ up to a new high point, then walks back down to $1$, and the next mountain climbs one step higher than the one before.
- So the numbers are far from random — they follow a strict up-then-down rule.
💡 Spotting the repeating climb-and-fall shape means you never have to read the terms one by one.
4.OA.C.5 Step 2 Check the square-numbered spots
- Rather than track every term, list only the positions that are perfect squares and read the value there straight from the given sequence.
- Position $1$ holds $1$, position $4$ holds $2$, position $9$ holds $3$, position $16$ holds $4$, position $25$ holds $5$.
- The value at a square position is exactly its square root: position $n^2$ holds the value $n$.
💡 Looking only at square positions turns a giant list into a tidy one-line rule.
5.NBT.B.5 Step 3 Is 2025 a perfect square?
- The rule only helps if $2025$ is a perfect square, so test it by multiplying.
- A number near $\sqrt{2000}$ is around $45$: try $44 \times 44 = 1936$ (too small) and $45 \times 45 = 2025$ (exact).
- So $2025 = 45^2$.
💡 A couple of well-aimed multiplications pin down the square root without any fancy tools.
4.OA.C.5 Step 4 Apply the rule and dodge the trap
- Since $2025 = 45^2$ is a perfect-square position, the rule gives value $45$.
- Watch the off-by-one trap: position $2025$ is still climbing; the mountain keeps rising to its peak of $46$ at position $2026$, and dropping to $44$ would be one step too far.
- Choice (D) $44$ is the bait for that slip, so the $2025$th term is $45$, which is $\textbf{(E)}$.
💡 The pattern names the answer directly; the only danger is drifting one seat left or right of position $n^2$.
4.OA.C.5 Read the sequence in blocks. It climbs $1, 2, 3, \dots$ up to a new high point, 4.OA.C.5 Rather than track every term, list only the positions that are perfect squares a 5.NBT.B.5 The rule only helps if $2025$ is a perfect square, so test it by multiplying. A 4.OA.C.5 Since $2025 = 45^2$ is a perfect-square position, the rule gives value $45$. Wat Review
Reasonableness: Cross-check by counting from a landmark. The mountains return to $1$ at positions $1, 3, 7, 13, 21, 31, \dots$, which are $n^2 - n + 1$. For $n = 45$ that is $45^2 - 45 + 1 = 1981$, a $1$. From there the sequence climbs $1, 2, 3, \dots$, so position $1981 + 44 = 2025$ holds $1 + 44 = 45$ — matching the square-position rule exactly. The peak $46$ only arrives at position $2026$, confirming $45$ and not $44$ or $46$.
Alternative: Tool #4 (Introduce a Variable): let the $n$th mountain's lowest point (a $1$) sit at position $n^2 - n + 1$, derived by summing the gaps $2, 4, 6, \dots$ between consecutive $1$s. Adding the climb of $n - 1$ steps lands the value $n$ at position $n^2$. This proves the rule algebraically instead of trusting the observed pattern, but it is more work than simply reading the square-position values off the list.
CCSS standards used (min grade 5)
4.OA.C.5Generate a number or shape pattern following a given rule (Recognizing the repeating climb-and-fall mountains and distilling the observation that the value at position $n^2$ is $n$, then applying that rule at position $2025$.)5.NBT.B.5Fluently multiply multi-digit whole numbers (Multiplying $44 \times 44$ and $45 \times 45$ to confirm that $2025 = 45^2$ is a perfect-square position.)
⭐ In a climbing-and-falling sequence, the term at a perfect-square position equals its square root — so position $45^2 = 2025$ simply holds $45$.
⭐ In a climbing-and-falling sequence, the term at a perfect-square position equals its square root — so position $45^2 = 2025$ simply holds $45$.
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