AMC 10 · 2025 · #5
Grade 5 patternConsider the sequence of positive integers
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…
What is the 2025th term in the sequence?
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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