Algebra · Chapter 15

Arithmetic Sequences

Add the same number again and again: find any term, count the terms, and sum them all with Gauss's pairing trick.

What's this chapter about?

15.0 Arithmetic Sequences
15.1 Practice Problems

You'll need

12.5 Setting up and combining equations

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15.00%

Arithmetic Sequences

In this section you will be able to find any term of an arithmetic sequence, count its terms, and add them all up fast.

Start at 1 and keep adding $k$ . For which $k$ do you ever land exactly on 2005? Try a few in the widget before you read on.

An arithmetic sequence adds the same number $d$ every step. That one fact gives you three tools: any term, the number of terms, and the sum.

Look: every question about a term is really a question about steps. From $a_1$ to $a_n$ is $n - 1$ steps of size $d$ .

For sums, pair the first term with the last. Every pair has the same total, so the average is just $\frac{\text{first} + \text{last}}{2}$ .

Exploreplay first, formulas later

Try this: Set d = 4. Is 2005 a term? Now try d = 5, 6 and 12. What do the good values of d have in common?
Try this: Raise "look at term k" one step at a time. How many steps of d does it take to get from a₁ to a_k?
Try this: Turn on "Show average and sum". Why does the dashed average line sit halfway between the first and last bars?
Try this: In the Sum card below, press Fold. How many columns are there, and how tall is each one?

Name it

Definition 15.0.1 · Arithmetic Sequence

An arithmetic sequence is a sequence of numbers with the same difference between consecutive terms.

1, 4, 7, 10, 13, \dots, 40

is an arithmetic sequence because there is always a difference of $3$ between consecutive terms.

Definition 15.0.3 · Arithmetic Sequence Notation

In general, the terms of an arithmetic sequence can be written as

a_1, a_2, a_3, a_4, \dots, a_n

where

$d$ is the common difference between consecutive terms
$n$ is the number of terms

Theorem 15.0.4 · $n$th Term in an Arithmetic Sequence

a_n = a_1 + (n - 1)d

a_n = a_m + (n - m)d

Theorem 15.0.5 · Number of Terms in an Arithmetic Sequence

n = \frac{a_n - a_1}{d} + 1

In words:

\text{Number of Terms} = \frac{\text{Last Term} - \text{First Term}}{\text{Common Difference}} + 1

Theorem 15.0.6 · Average of Terms in an Arithmetic Sequence

\text{Average of Terms} = \frac{a_1 + a_n}{2}

Theorem 15.0.7 · Sum of All Terms in an Arithmetic Sequence

S_n = \frac{a_1 + a_n}{2} \times n

Substitute $a_n = a_1 + (n - 1)d$ to get

S_n = \frac{2a_1 + (n - 1)d}{2} \times n

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Example 15.1

AIME

For each positive integer $k$ , let $S_k$ denote the increasing arithmetic sequence of integers whose first term is $1$ and whose common difference is $k$ . For example, $S_3$ is the sequence $1, 4, 7, 10, \dots$ . For how many values of $k$ does $S_k$ contain the term $2005$ ?

1
What is the $m$ th term of $S_k$ ?

Predict first:
Click a choice or type its letter (A–E).

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Example 15.2

AIME

If the integer $k$ is added to each of the numbers $36$ , $300$ , and $596$ , one obtains the squares of three consecutive terms of an arithmetic series. Find $k$ .

1
Call the three consecutive terms $a - d$ , $a$ , $a + d$ . What three equations do you get?

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Problem 15.1.5

AMC 10

A grocer makes a display of cans in which the top row has one can and each lower row has two more cans than the row above it. If the display contains $100$ cans, how many rows does it contain?

Problem 15.1.1

AMC 10

What is the greatest number of consecutive integers whose sum is $45$ ?

Problem 15.1.7

AMC 10

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 $a_2 - a_1$ ?

Contest cornerreal contest problems

Problem 15.1.2

AMC 10

In how many ways can $345$ be written as the sum of an increasing sequence of two or more consecutive positive integers?

Problem 15.1.4

AMC 10

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?

Problem 15.1.6

AIME

The terms of an arithmetic sequence add to $715$ . The first term of the sequence is increased by $1$ , the second term is increased by $3$ , the third term is increased by $5$ , and in general, the $k$ th term is increased by the $k$ th odd positive integer. The terms of the new sequence add to $836$ . Find the sum of the first, last, and middle terms of the original sequence.

Problem 15.1.8(additional)

AMC 10

Mary divides a circle into $12$ sectors. The central angles of these sectors, measured in degrees, are all integers and they form an arithmetic sequence. What is the degree measure of the smallest possible sector angle?

Problem 15.1.10(additional)

AIME

The degree measures of the angles in a convex $18$ -sided polygon form an increasing arithmetic sequence with integer values. Find the degree measure of the smallest angle.

Checkpoint for section 15.0

Get 2 of 3 right to master this section. One try each, no hints.

15.1

Practice Problems

In this section you will be able to mix arithmetic sequences with averages, polynomials and hidden quadratics.

These problems hide the arithmetic sequence: inside a set’s averages, a triangle’s angles, a polynomial’s roots, or a product of two sequences. Every problem has three hints if you get stuck.

Want every problem from this chapter on one page? Open the problem set.

Mixed practice

Problem 15.1.11(additional)

Find the roots of the polynomial $x^5 - 5x^4 - 35x^3 + ax^2 + bx + c$ , given that the roots form an arithmetic progression.

Contest cornerreal contest problems

Problem 15.1.3

AMC 10/12

Suppose that $S$ is a finite set of positive integers. If the greatest integer in $S$ is removed from $S$ , then the average value (arithmetic mean) of the integers remaining is $32$ . If the least integer in $S$ is also removed, then the average value of the integers remaining is $35$ . If the greatest integer is then returned to the set, the average value of the integers rises to $40$ . The greatest integer in the original set $S$ is $72$ greater than the least integer in $S$ . What is the average value of all the integers in the set $S$ ?

Problem 15.1.9(additional)

AMC 10

How many non-similar triangles have angles whose degree measures are distinct positive integers in arithmetic progression?

Problem 15.1.12(additional)

AIMEChallenge

Find the eighth term of the sequence $1440, 1716, 1848, \ldots$ , whose terms are formed by multiplying the corresponding terms of two arithmetic sequences.