Sequences and series | AMC Intermediate topics (AMC 10/12)

This chapter only applies to AMC 10/12 test takers.

A sequence is an ordered list of numbers that follows a pattern. A term is a specific number in that list.

When you add the terms of a sequence, you get a series. Many AMC problems focus on two common types of sequences - arithmetic and geometric - and on summing their terms.

Arithmetic sequences

$a_{1} = a$

$a_{2} = a + c$

$a_{3} = a + 2 c$

$a_{4} = a + 3 c$

$\dots$

$a_{n} = a_{n - 1} + c$

$OR$

$a_{n} = a_{1} + (n - 1) c$

An arithmetic sequence is a sequence where each term is found by adding the same constant to the previous term. This constant is called the common difference (often written as $c$ ). The common difference can be positive or negative.

Example with common difference $3$ : $1, 4, 7, 10, 13$
Example with common difference $- 3$ : $13, 10, 7, 4, 1$

You can think of the common difference as the difference between consecutive terms.

Mean and sum of an arithmetic sequence

Because the terms are equally spaced, the average (arithmetic mean) of the terms is the average of the first and last terms.

To find the sum of the terms, multiply:

(average of the terms) × (number of terms)

Finding a specific term

To find a particular term $a_{n}$ , you can use either of the formulas shown in the block above.

Example: The question below is from 2002 AMC 10B

Suppose that ${a_{n}}$ is an arithmetic sequence with

$a_{1} + a_{2} + \dots + a_{100} = 100 and a_{101} + a_{102} + \dots + a_{200} = 200.$

What is the value of $a_{2} - a_{1} ?$ A. $0.0001$
B. $0.001$
C. $0.01$
D. $0.1$
E. $1$

(spoiler)

Answer: C. 0.01

The sum of the first $100$ terms is $100$ , so the average of those $100$ equally spaced terms is $100/100 = 1$ . In an arithmetic sequence, the average of a block of equally spaced terms is also the value of the middle (median) term(s), so the “center” of terms $a_{1}$ through $a_{100}$ is at $1$ .

The sum of terms $a_{101}$ through $a_{200}$ is $200$ , so the average of those $100$ terms is $200/100 = 2$ . The center of that block is at $2$ .

The centers of these two blocks are $1$ apart (from $1$ to $2$ ). Moving from the center of the first block to the center of the second block advances by $100$ terms, so the common difference is

$1/100 = 0.01$

Since $a_{2} - a_{1}$ is exactly the common difference, $a_{2} - a_{1} = 0.01$ .

Geometric sequences

$a_{1} = a$

$a_{2} = a r$

$a_{3} = a r^{2}$

$\dots$

$a_{n} = a_{n - 1} (r)$

$OR$

$a_{n} = a r^{n - 1}$

A geometric sequence is similar to an arithmetic sequence, but instead of adding a constant, you multiply by a constant ratio $r$ to get the next term.

If $a_{1} = 1$ and $r = 3$ , the sequence begins $1, 3, 9, 27, \dots$
If $r$ is negative, the signs alternate.

To find a specific term $a_{n}$ , you can use either of the formulas shown in the block above.

Example: The question below is from 2024 AMC 10A

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 $b ?$
A. $9$
B. $12$
C. $16$
D. $18$
E. $21$

(spoiler)

Answer: E. 21

Let the common ratio be $r$ . Then $720 = a r$ and $b = 720 r$ .

To make $b$ as small as possible while keeping $a$ and $b$ integers with $a < 720 < b$ , we want the smallest rational $r > 1$ such that:

$720 r$ is an integer, and
$720/ r$ is an integer.

The smallest such ratio is $16/15$ .

$b = 720 (16/15) = 768$

$7 + 6 + 8 = 21$

Series

A series is the sum of the terms in a sequence. Here is the summation notation commonly used for series.

$S_{n} = k = 1 \sum n a_{k}$

The notation $S_{n}$ means “the sum of the first $n$ terms.” Here, $a_{k}$ is the value of the $k$ th term.

Special series

Some summations come up so often that it helps to know their closed-form formulas.

Sum of squares

$k = 1 \sum n k^{2} = 1^{2} + 2^{2} + 3^{2} + \dots + n^{2} = \frac{n ( n + 1 ) ( 2 n + 1 )}{6}$

Sum of cubes

$k = 1 \sum n k^{3} = 1^{3} + 2^{3} + 3^{3} + \dots + n^{3} = (\frac{n ( n + 1 )}{2})^{2}$

Common themes

The sum of the first $x$ odd numbers equals $x^{2}$ .
Make sure you are careful to recognize when a question asks for either $a_{n}$ or $S_{n}$ . The first is simply the nth term, the second is the sum of the terms of the sequence up to the nth term.
Recognize when you can use telescoping to collapse and simplify a sum. In other words, be on the lookout for the opportunity to simplify sums by finding where terms cancel each other out.
You will oftentimes find trap answers if you miscount by one in the sequence. Also, be very careful not to multiply the ratio or add the difference in the very first term of the sequence. This will lead to an incorrect solution.

This chapter only applies to AMC 10/12 test takers.

A sequence is an ordered list of numbers that follows a pattern. A term is a specific number in that list.

When you add the terms of a sequence, you get a series. Many AMC problems focus on two common types of sequences - arithmetic and geometric - and on summing their terms.

Arithmetic sequences

$a_{1} = a$

$a_{2} = a + c$

$a_{3} = a + 2 c$

$a_{4} = a + 3 c$

$\dots$

$a_{n} = a_{n - 1} + c$

$OR$

$a_{n} = a_{1} + (n - 1) c$

Example with common difference $3$ : $1, 4, 7, 10, 13$
Example with common difference $- 3$ : $13, 10, 7, 4, 1$

You can think of the common difference as the difference between consecutive terms.

Mean and sum of an arithmetic sequence

Because the terms are equally spaced, the average (arithmetic mean) of the terms is the average of the first and last terms.

To find the sum of the terms, multiply:

(average of the terms) × (number of terms)

Finding a specific term

To find a particular term $a_{n}$ , you can use either of the formulas shown in the block above.

Example: The question below is from 2002 AMC 10B

Suppose that ${a_{n}}$ is an arithmetic sequence with

$a_{1} + a_{2} + \dots + a_{100} = 100 and a_{101} + a_{102} + \dots + a_{200} = 200.$

What is the value of $a_{2} - a_{1} ?$ A. $0.0001$
B. $0.001$
C. $0.01$
D. $0.1$
E. $1$

(spoiler)

Answer: C. 0.01

The sum of terms $a_{101}$ through $a_{200}$ is $200$ , so the average of those $100$ terms is $200/100 = 2$ . The center of that block is at $2$ .

The centers of these two blocks are $1$ apart (from $1$ to $2$ ). Moving from the center of the first block to the center of the second block advances by $100$ terms, so the common difference is

$1/100 = 0.01$

Since $a_{2} - a_{1}$ is exactly the common difference, $a_{2} - a_{1} = 0.01$ .

Geometric sequences

$a_{1} = a$

$a_{2} = a r$

$a_{3} = a r^{2}$

$\dots$

$a_{n} = a_{n - 1} (r)$

$OR$

$a_{n} = a r^{n - 1}$

A geometric sequence is similar to an arithmetic sequence, but instead of adding a constant, you multiply by a constant ratio $r$ to get the next term.

If $a_{1} = 1$ and $r = 3$ , the sequence begins $1, 3, 9, 27, \dots$
If $r$ is negative, the signs alternate.

To find a specific term $a_{n}$ , you can use either of the formulas shown in the block above.

Example: The question below is from 2024 AMC 10A

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 $b ?$
A. $9$
B. $12$
C. $16$
D. $18$
E. $21$

(spoiler)

Answer: E. 21

Let the common ratio be $r$ . Then $720 = a r$ and $b = 720 r$ .

To make $b$ as small as possible while keeping $a$ and $b$ integers with $a < 720 < b$ , we want the smallest rational $r > 1$ such that:

$720 r$ is an integer, and
$720/ r$ is an integer.

The smallest such ratio is $16/15$ .

$b = 720 (16/15) = 768$

$7 + 6 + 8 = 21$

Series

A series is the sum of the terms in a sequence. Here is the summation notation commonly used for series.

$S_{n} = k = 1 \sum n a_{k}$

The notation $S_{n}$ means “the sum of the first $n$ terms.” Here, $a_{k}$ is the value of the $k$ th term.

Special series

Some summations come up so often that it helps to know their closed-form formulas.

Sum of squares

$k = 1 \sum n k^{2} = 1^{2} + 2^{2} + 3^{2} + \dots + n^{2} = \frac{n ( n + 1 ) ( 2 n + 1 )}{6}$

Sum of cubes

$k = 1 \sum n k^{3} = 1^{3} + 2^{3} + 3^{3} + \dots + n^{3} = (\frac{n ( n + 1 )}{2})^{2}$

Common themes

The sum of the first $x$ odd numbers equals $x^{2}$ .
Make sure you are careful to recognize when a question asks for either $a_{n}$ or $S_{n}$ . The first is simply the nth term, the second is the sum of the terms of the sequence up to the nth term.
Recognize when you can use telescoping to collapse and simplify a sum. In other words, be on the lookout for the opportunity to simplify sums by finding where terms cancel each other out.
You will oftentimes find trap answers if you miscount by one in the sequence. Also, be very careful not to multiply the ratio or add the difference in the very first term of the sequence. This will lead to an incorrect solution.