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Sequences & Series

Arithmetic/geometric sequences, recursions, sums, telescoping sums

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Sequences & SeriesAMC 10 Step by Step

Name: Date: October 28, 2026

20 problems, easiest first

What you need to know

We write $a_n$ for the $n$ th term of a sequence, so $a_1$ is the first term.

Arithmetic sequences go up by the same amount $d$ each time (the common difference). The $n$ th term is $a_n=a_1+(n-1)d$ . The sum of the first $n$ terms is $S_n=\dfrac n2(a_1+a_n)$ : the number of terms times the average of the first and last. Example: for $5,8,11,\dots$ , $a_{20}=5+19\cdot3=62$ and $S_{20}=20\cdot\frac{5+62}{2}=670$ .
Geometric sequences multiply by the same number $r$ each time (the common ratio). Then $a_n=a_1r^{n-1}$ and $S_n=a_1\dfrac{r^n-1}{r-1}$ . Example: for $3,6,12,\dots$ , $a_6=3\cdot2^5=96$ and $S_6=3\cdot\frac{2^6-1}{2-1}=189$ .
Infinite geometric series. If $r$ is between $-1$ and $1$ , the sum that goes on forever is $\dfrac{a_1}{1-r}$ . Example: $8+4+2+1+\dots=\frac{8}{1-\frac12}=16$ .
Sums to memorize. $1+2+\dots+n=\dfrac{n(n+1)}2$ . $1^2+2^2+\dots+n^2=\dfrac{n(n+1)(2n+1)}6$ . $1^3+2^3+\dots+n^3=\left(\dfrac{n(n+1)}2\right)^2$ . Example: with $n=10$ these are $55$ , $385$ , and $55^2=3025$ .
Telescoping. Write each term as a difference so that neighbors cancel and only the ends survive. The key split is $\dfrac1{k(k+1)}=\dfrac1k-\dfrac1{k+1}$ (this is partial fractions). Example: $\frac1{1\cdot2}+\frac1{2\cdot3}+\dots+\frac1{99\cdot100}=1-\frac1{100}=\frac{99}{100}$ .
Recursions often repeat. When each term is made from the one before, compute terms until one repeats. From there the sequence cycles. If the cycle goes all the way back to $a_1$ , only the remainder of the index divided by the cycle length (the period) matters. If some terms come before the cycle starts, count from the start of the cycle instead: adding the squares of the digits, starting from $2$ , gives $2, 4, 16, 37, 58, 89, 145, 42, 20, 4, \dots$ , a cycle of length $8$ that starts at $a_2$ . Example: $a_1=2$ and $a_{n+1}=\frac1{1-a_n}$ give $2,-1,\frac12,2,\dots$ , with period $3$ . Since $100=3\cdot33+1$ , $a_{100}=a_1=2$ .
Three-term tests. $a,b,c$ is arithmetic exactly when $2b=a+c$ . For numbers that are not zero, $a,b,c$ is geometric exactly when $b^2=ac$ . Example: $4,7,10$ is arithmetic ( $14=4+10$ ) and $4,6,9$ is geometric ( $36=4\cdot9$ ).

How AMC 10 tests it

Problems 3–10: a direct question about an arithmetic sequence (one term, a sum, or the number of terms).
Problems 8–15: " $a_1=\dots$ , $a_{n+1}=\dots$ ; find $a_{2025}$ ." Compute a few terms, spot the cycle, and use the remainder of the index.
Problems 10–18: an arithmetic or geometric sequence hidden in a story (rows of seats, a bouncing ball), sometimes an infinite geometric series.
Problems 14–20: a telescoping sum of fractions, or of terms like $k\cdot k!$ . Rewrite each term as a difference, such as $k\cdot k!=(k+1)!-k!$ .
Mixed conditions (" $a,b,c$ is arithmetic and $a,c,b$ is geometric") that turn into a small system of equations.

Standard approaches

Decide the type: arithmetic, geometric, telescoping, or recursive. For a recursion, write out the first five or six terms.
For an arithmetic sum, use (number of terms) $\times$ (average term). Count the terms as $\frac{\text{last}-\text{first}}d+1$ .
For a sum of fractions, or a product, try to write each term as a difference of neighboring pieces, like $\frac1k-\frac1{k+1}$ .
For a term with a huge index, find the period or a formula, then reduce the index.
Use the answer choices as a check: whole number or not, even or odd, or the pattern from small cases.

Worked example

Define $a_1=3$ and $a_{n+1}=\dfrac{a_n-1}{a_n+1}$ for $n\ge1$ . What is the product $a_1a_2a_3\cdots a_{2026}$ ?

(A) $-2$ (B) $-\frac13$ (C) $\frac12$ (D) $1$ (E) $\frac32$

Compute terms until one repeats.
$a_2=\frac{3-1}{3+1}=\frac12,\quad a_3=\frac{\frac12-1}{\frac12+1}=\frac{-\frac12}{\frac32}=-\frac13,\quad a_4=\frac{-\frac13-1}{-\frac13+1}=\frac{-\frac43}{\frac23}=-2,\quad a_5=\frac{-2-1}{-2+1}=3.$
Since $a_5=a_1$ , the terms repeat in blocks of $4$ : $3,\frac12,-\frac13,-2$ .

Multiply one block. $3\cdot\frac12\cdot\left(-\frac13\right)\cdot(-2)=1$ .

Count the blocks. $2026=4\cdot506+2$ . So the $2026$ terms are $506$ full blocks, each multiplying to $1$ , then two extra terms: $a_{2025}=a_1=3$ and $a_{2026}=a_2=\frac12$ .

The whole product is $1\cdot3\cdot\frac12=\frac32$ . The answer is $\boxed{\textbf{(E)}\ \frac32}$ .

Pitfalls

Off-by-one when counting terms: $a_n$ is $n-1$ steps past $a_1$ . Example: for $5,8,11,\dots$ , $a_{20}=5+19\cdot3$ , not $5+20\cdot3$ .
Using $\frac{a_1}{1-r}$ when $r$ is not between $-1$ and $1$ . Example: $1+2+4+\dots$ has no sum, but the formula would give $-1$ .
Reducing the index wrongly. With period $4$ and a cycle that starts at $a_1$ , an index that is a multiple of $4$ (remainder $0$ , like $2024$ ) matches $a_4$ , not a nonexistent $a_0$ .
Telescoping without tracking which terms survive at both ends, or dropping the factor in front: $\dfrac1{k(k+2)}=\dfrac12\left(\dfrac1k-\dfrac1{k+2}\right)$ , not $\dfrac1k-\dfrac1{k+2}$ .

Traps that recur

Trying to compute terms exactly, or bounding with only one side of the ratio and landing in the wrong interval. (2019 AMC 10B #24)
Reading 'the average drops by 1' as 'each term drops by 1' and running the sequence down from around 2025 to 1, which points at choice (E). (2025 AMC 10B #17)
Assuming the ratio must be an integer (b = 1440, digit sum 9) or grabbing 9/8 (b = 810) without checking that 15 and 16 both divide 720. (2024 AMC 10A #19)
Assuming every row has the same common difference (a linear function Bi + Cj + A), which cannot fit the four given values; the row differences themselves form an arithmetic progression. (2024 AMC 10A #21)

1.2011 AMC 10A #17
In the eight-term sequence $A,B,C,D,E,F,G,H$ , the value of $C$ is 5 and the sum of any three consecutive terms is 30. What is $A+H$ ?
1. A) $17$
2. B) $18$
3. C) $25$
4. D) $26$
5. E) $43$
2.2009 AMC 10A #15
The figures $F_1$ , $F_2$ , $F_3$ , and $F_4$ shown are the first in a sequence of figures. For $n\ge3$ , $F_n$ is constructed from $F_{n - 1}$ by surrounding it with a square and placing one more diamond on each side of the new square than $F_{n - 1}$ had on each side of its outside square. For example, figure $F_3$ has $13$ diamonds. How many diamonds are there in figure $F_{20}$ ?
1. A) $401$
2. B) $485$
3. C) $585$
4. D) $626$
5. E) $761$
3.2009 AMC 10B #14
On Monday, Millie puts a quart of seeds, $25\%$ of which are millet, into a bird feeder. On each successive day she adds another quart of the same mix of seeds without removing any seeds that are left. Each day the birds eat only $25\%$ of the millet in the feeder, but they eat all of the other seeds. On which day, just after Millie has placed the seeds, will the birds find that more than half the seeds in the feeder are millet?
1. A) $\text{Tuesday}$
2. B) $\text{Wednesday}$
3. C) $\text{Thursday}$
4. D) $\text{Friday}$
5. E) $\text{Saturday}$
4.2006 AMC 10A #14
A number of linked rings, each $1$ cm thick, are hanging on a peg. The top ring has an outside diameter of $20$ cm. The outside diameter of each of the outer rings is $1$ cm less than that of the ring above it. The bottom ring has an outside diameter of $3$ cm. What is the distance, in cm, from the top of the top ring to the bottom of the bottom ring?
1. A) $171$
2. B) $173$
3. C) $182$
4. D) $188$
5. E) $210$
5.2006 AMC 10B #18
Let $a_1 , a_2 , ...$ be a sequence for which $a_1=2$ , $a_2=3$ , and $a_n=\frac{a_{n-1}}{a_{n-2}}$ for each positive integer $n \ge 3$ . What is $a_{2006}$ ?
1. A) $\frac{1}{2}$
2. B) $\frac{2}{3}$
3. C) $\frac{3}{2}$
4. D) $2$
5. E) $3$
6.2005 AMC 10A #17
In the five-sided star shown, the letters $A$ , $B$ , $C$ , $D$ , and $E$ are replaced by the numbers $3$ , $5$ , $6$ , $7$ , and $9$ , although not necessarily in this order. The sums of the numbers at the ends of the line segments $\overline{AB}$ , $\overline{BC}$ , $\overline{CD}$ , $\overline{DE}$ , and $\overline{EA}$ form an arithmetic sequence, although not necessarily in this order. What is the middle term of the arithmetic sequence?
1. A) $9$
2. B) $10$
3. C) $11$
4. D) $12$
5. E) $13$
7.2005 AMC 10B #11
The first term of a sequence is $2005$ . Each succeeding term is the sum of the cubes of the digits of the previous term. What is the ${2005}^{\text{th}}$ term of the sequence?
1. A) $29$
2. B) $55$
3. C) $85$
4. D) $133$
5. E) $250$
8.2004 AMC 10A #18
A sequence of three real numbers forms an arithmetic progression with a first term of $9$ . If $2$ is added to the second term and $20$ is added to the third term, the three resulting numbers form a geometric progression. What is the smallest possible value for the third term in the geometric progression?
1. A) $1$
2. B) $4$
3. C) $36$
4. D) $49$
5. E) $81$
9.2004 AMC 10B #19
In the sequence $2001$ , $2002$ , $2003$ , $\ldots$ , each term after the third is found by subtracting the previous term from the sum of the two terms that precede that term. For example, the fourth term is $2001 + 2002 - 2003 = 2000$ . What is the $2004^\textrm{th}$ term in this sequence?
1. A) $-2004$
2. B) $-2$
3. C) $0$
4. D) $4003$
5. E) $6007$
10.2002 AMC 10B #19
Suppose that $\{a_n\}$ is an arithmetic sequence with $a_1+a_2+\cdots+a_{100}=100 \text{ and } a_{101}+a_{102}+\cdots+a_{200}=200.$ What is the value of $a_2 - a_1 ?$
1. A) $0.0001$
2. B) $0.001$
3. C) $0.01$
4. D) $0.1$
5. E) $1$
11.2025 AMC 10B #17
Consider a decreasing sequence of $n$ positive integers $x_1 > x_2 > \dotsb > x_n$ that satisfies the following two conditions:
$\qquad\bullet$ The average (arithmetic mean) of the first $3$ terms in the sequence is $2025.$
$\qquad\bullet$ For all $4 \leq k \leq n,$ the average of the first $k$ terms in the sequence is $1$ less than the average of the first $k - 1$ terms in the sequence.
What is the greatest possible value of ${n?}$
1. A) $1013$
2. B) $1014$
3. C) $1016$
4. D) $2016$
5. E) $2025$
12.2024 AMC 10A #19
The first three terms of a geometric sequence are the integers $a, 720$ and $b$ , where $a < 720 < b$ . What is the sum of the digits of the least possible value of $b$ ?
1. A) $9$
2. B) $12$
3. C) $16$
4. D) $18$
5. E) $21$
13.2024 AMC 10A #21
The numbers, in order, of each row and the numbers, in order, of each column of a $5 \times 5$ array of integers form an arithmetic progression of length $5$ . The numbers in positions $(5, 5)$ , $(2, 4)$ , $(4, 3)$ and $(3, 1)$ are $0$ , $48$ , $16$ , and $12$ , respectively. What number is in position $(1, 2)$ ? $\begin{bmatrix} . & ? &.&.&. \\ .&.&.&48&.\\ 12&.&.&.&.\\ .&.&16&.&.\\ .&.&.&.&0\end{bmatrix}$
1. A) $19$
2. B) $24$
3. C) $29$
4. D) $34$
5. E) $39$
14.2024 AMC 10B #23
The Fibonacci numbers are defined by $F_1 = 1, F_2 = 1,$ and $F_n = F_{n-1} + F_{n-2}$ for $n \geq 3.$ What is ${\frac{F_2}{F_1}} + {\frac{F_4}{F_2}} + {\frac{F_6}{F_3}} + ... + {\frac{F_{20}}{F_{10}}}?$
1. A) $318$
2. B) $319$
3. C) $320$
4. D) $321$
5. E) $322$
15.2023 AMC 10B #23
An arithmetic sequence of positive integers has $n\ge3$ terms, initial term $a$ , and common difference $d>1$ . Carl wrote down all the terms in this sequence correctly except for one term, which was off by $1$ . The sum of the terms he wrote down was $222$ . What is $a+d+n$ ?
1. A) $24$
2. B) $20$
3. C) $22$
4. D) $28$
5. E) $26$
16.2022 AMC 10A #20
A four-term sequence is formed by adding each term of a four-term arithmetic sequence of positive integers to the corresponding term of a four-term geometric sequence of positive integers. The first three terms of the resulting four-term sequence are $57$ , $60$ , and $91$ . What is the fourth term of this sequence?
1. A) $190$
2. B) $194$
3. C) $198$
4. D) $202$
5. E) $206$
17.2019 AMC 10A #23
Travis has to babysit the terrible Thompson triplets. Knowing that they love big numbers, Travis devises a counting game for them. First Tadd will say the number $1$ , then Todd must say the next two numbers ( $2$ and $3$ ), then Tucker must say the next three numbers ( $4$ , $5$ , $6$ ), then Tadd must say the next four numbers ( $7$ , $8$ , $9$ , $10$ ), and the process continues to rotate through the three children in order, each saying one more number than the previous child did, until the number $10,000$ is reached. What is the $2019$ th number said by Tadd?
1. A) $5743$
2. B) $5885$
3. C) $5979$
4. D) $6001$
5. E) $6011$
18.2015 AMC 10B #24
Aaron the ant walks on the coordinate plane according to the following rules. He starts at the origin $p_0=(0,0)$ facing to the east and walks one unit, arriving at $p_1=(1,0)$ . For $n=1,2,3,\dots$ , right after arriving at the point $p_n$ , if Aaron can turn $90^\circ$ left and walk one unit to an unvisited point $p_{n+1}$ , he does that. Otherwise, he walks one unit straight ahead to reach $p_{n+1}$ . Thus the sequence of points continues $p_2=(1,1), p_3=(0,1), p_4=(-1,1), p_5=(-1,0)$ , and so on in a counterclockwise spiral pattern. What is $p_{2015}$ ?
1. A) $(-22,-13)$
2. B) $(-13,-22)$
3. C) $(-13,22)$
4. D) $(13,-22)$
5. E) $(22,-13)$
19.2014 AMC 10A #24
A sequence of natural numbers is constructed by listing the first $4$ , then skipping one, listing the next $5$ , skipping $2$ , listing $6$ , skipping $3$ , and, on the $n$ th iteration, listing $n+3$ and skipping $n$ . The sequence begins $1,2,3,4,6,7,8,9,10,13$ . What is the $500,000^{\text{th}}$ number in the sequence?
1. A) $996,506$
2. B) $996,507$
3. C) $996,508$
4. D) $996,509$
5. E) $996,510$
20.2013 AMC 10B #21
Two non-decreasing sequences of nonnegative integers have different first terms. Each sequence has the property that each term, beginning with the third, is the sum of the previous two terms, and the seventh term of each sequence is $N$ . What is the smallest possible value of $N$ ?
1. A) $55$
2. B) $89$
3. C) $104$
4. D) $144$
5. E) $273$

Sequences & Series — answersAMC 10 Step by Step

Name: Date: October 28, 2026

Keep this page separate from the problems.

Answers

#	Answer	The idea it turns on
1	C	Two overlapping groups of three share two terms, so each term equals the term three places later. So H equals B, and A + B = 30 − 5.
2	E	Each new ring with n diamonds per side holds 4(n - 1) diamonds, so F_20 = 1 + 4(1 + 2 + ... + 19) = 761.
3	D	After each refill there is always exactly 3/4 quart of other seeds, since the birds eat them all. Track only the millet until it passes 3/4 quart.
4	B	Each ring hangs on the one above, so the holes line up end to end. Add the holes, 18 down to 1, plus 1 cm at each end.
5	E	Write out terms until 2 and 3 appear again. The sequence repeats every 6 terms, and 2006 leaves remainder 2, so use the second term.
6	D	Each letter is on exactly two segments, so the five sums add to twice 30, which is 60. The middle of five evenly spaced numbers is their average, 12.
7	E	Write out terms until one repeats. From term 2 on, the sequence cycles 133, 55, 250 every 3 terms, and term 2005 lands on 250, like term 4.
8	A	The new numbers are 9, 11 + d and 29 + 2d. Making them geometric gives d = 10 or d = -14; the second gives third term 1.
9	C	List the first ten terms. The even-numbered terms drop by 2 each time: 2002, 2000, 1998, ... After 1001 drops, term 2004 is 0.
10	C	Each term in the second block is 100 steps after its partner in the first, so it is 100d bigger. The sums differ by 10000d = 100.
11	B	The average of the first k terms is 2028 − k, so their total is k(2028 − k). Subtracting totals shows later terms drop by 2.
12	E	The middle term squared equals the product of the outer terms, so ab = 720². Make a the largest divisor of 720² below 720.
13	C	Measure from the bottom-right 0. Column 5 goes up in equal steps, and the row steps themselves change by equal amounts, so each entry is P·u + v·(Q + R·u).
14	B	The ten fractions come out to whole numbers 1, 3, 4, 7, 11, ... and each one is the sum of the two before it, just like Fibonacci numbers.
15	B	The real sum is 221 or 223. The number of terms divides twice the sum and is at most 14, since steps are at least 2. Only 13 fits.
16	E	Subtracting neighboring terms twice removes the arithmetic part and leaves b(r - 1) squared = 28. So the ratio r is 2 or 3.
17	C	Tadd speaks in every third turn, saying 1, 4, 7, ... numbers. Add these to find which turn holds his 2019th number, then find where that turn starts.
18	D	At square step numbers the ant sits at a spiral corner. Step 1936 is at (-22, 22); then 44 steps down and 35 steps right.
19	A	The 500,000th listed number is 500,000 plus how many numbers were skipped before it. Up to that point, 1 + 2 + ... + 996 = 496,506 numbers were skipped.
20	C	The seventh term is 5a + 8b. Two different starts give the same seventh term only if one start is 8 more in a and 5 less in b.

Problems © Mathematical Association of America (MAA), American Mathematics Competitions. Reproduced for non-commercial educational use. The topic note, tags and key insights are original to this site.