Algebra 2 / Precalculus · Step-by-step guide

Arithmetic Sequences

Find terms and sums of arithmetic sequences using common differences and explicit formulas.

By Tyler Blovat··9 min read
The short answer

Start with the central idea

An arithmetic sequence adds the same common difference d each time. Its nth term is an=a1+(n−1)d\displaystyle a_n=a_1+(n-1)d, and the first n terms sum to Sn=n(a1+an)/2\displaystyle S_n=n(a_1+a_n)/2.

an=a1+(n−1)d\displaystyle a_n=a_1+(n-1)d
01

How it works

Build the method from meaning before memorizing the moves.

These are the decisions that make the procedure reliable. Read them once, then look for each idea in the worked examples below.

1

Look for equal differences

Subtract consecutive terms. A constant result identifies an arithmetic sequence.

2

Use n − 1 intervals

From the first term to the nth term there are n − 1 equal jumps.

3

Average the endpoints

The arithmetic-series sum equals number of terms times the average of first and last terms.

02

See the structure

A picture makes the relationships easier to remember.

123456n
A sequence is discrete: each plotted point corresponds to an integer term number n.
03

Worked examples

Follow the reason for each line, then try to reproduce it without looking.

Example 1Find a term

Find the 20th term of 5, 8, 11, …

  1. a1=5\displaystyle a_1=5 and d=3\displaystyle d=3.
  2. a20=5+(20−1)3\displaystyle a_{20}=5+(20-1)3.
  3. Compute 19 equal jumps.
Answer62
Example 2Build the formula

Sequence 12, 7, 2, …

  1. The common difference is −5.
  2. Use an=12+(n−1)(−5)\displaystyle a_n=12+(n-1)(-5).
  3. Simplify if desired.
Answeran=17−5n\displaystyle a_n=17-5n
Example 3Find a sum

Sum the first 10 terms of 4, 7, 10, …

  1. a10=4+9(3)=31\displaystyle a_{10}=4+9(3)=31.
  2. S10=10(4+31)/2\displaystyle S_{10}=10(4+31)/2.
  3. Evaluate.
Answer175
04

Common mistakes—and how to avoid them

Accuracy improves fastest when you know what to check.

  • Using n instead of n − 1.

  • Confusing a common difference with a common ratio.

  • Using the sum formula without finding the correct last term.

05

Turn the explanation into a skill

Reading creates recognition. Independent practice creates recall.

Free printable worksheet · Answer key included

Sequences and Series

Practice arithmetic and geometric patterns, formulas, and sums.

Preview the worksheet

A strong practice loop: solve one example with the guide open, solve a similar problem without it, explain the method aloud, and return the next day for a short mixed review.

06

Practice problems with worked solutions

Solve each problem on paper, use the hint only if needed, then compare every step.

1

Find the 10th term of 4, 7, 10, …

Show hint

Use an=a1+(n−1)d\displaystyle a_n=a_1+(n-1)d.

Show worked solution
  1. a1=4\displaystyle a_1=4 and d=3\displaystyle d=3.
  2. a10=4+9(3)=31\displaystyle a_{10}=4+9(3)=31.

Answer: 31\displaystyle 31

2

Find d when a3=11\displaystyle a_3=11 and a8=31\displaystyle a_8=31.

Show hint

Five steps separate the terms.

Show worked solution
  1. a8−a3=5d\displaystyle a_8-a_3=5d.
  2. 31−11=20=5d\displaystyle 31-11=20=5d.
  3. d=4\displaystyle d=4.

Answer: 4\displaystyle 4

3

Write an explicit formula for 12, 7, 2, …

Show hint

The common difference is −5.

Show worked solution
  1. a1=12,d=−5\displaystyle a_1=12,d=-5.
  2. an=12+(n−1)(−5)\displaystyle a_n=12+(n-1)(-5).

Answer: an=12−5(n−1)\displaystyle a_n=12-5(n-1)

4

Find the sum of the first 20 terms of 2, 5, 8, …

Show hint

Find the last term, then use the sum formula.

Show worked solution
  1. a20=2+19(3)=59\displaystyle a_{20}=2+19(3)=59.
  2. S20=20(2+59)/2\displaystyle S_{20}=20(2+59)/2.
  3. S20=610\displaystyle S_{20}=610.

Answer: 610\displaystyle 610

5

Which term of 6, 10, 14, … equals 50?

Show hint

Set the explicit formula equal to 50.

Show worked solution
  1. 50=6+(n−1)4\displaystyle 50=6+(n-1)4.
  2. 44=4(n−1)\displaystyle 44=4(n-1), so n−1=11\displaystyle n-1=11.
  3. n=12\displaystyle n=12.

Answer: The 12th term.

07

Frequently asked questions

Quick answers before you move on.

What should I remember first?

An arithmetic sequence adds the same common difference d each time. Its nth term is an=a1+(n−1)d\displaystyle a_n=a_1+(n-1)d, and the first n terms sum to Sn=n(a1+an)/2\displaystyle S_n=n(a_1+a_n)/2.

How do I know whether I understand this topic?

You should be able to name the method, explain why each step is valid, complete a new example without copying, and check whether your answer is reasonable.

What should I do if I keep making the same mistake?

Write the error as a specific checkpoint—for example, “I will identify the hypotenuse before substituting.” Then use that checkpoint on three short problems rather than repeating a full page without feedback.

Keep learning

Related guides