Algebra 1 Arithmetic and Geometric Sequences Worksheets with Answers (Free PDF)

Arithmetic and geometric sequences worksheet – Maths PDF

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This arithmetic and geometric sequences worksheet for Algebra 1 covers arithmetic and geometric sequences (Common Core F-IF.A.3, F-BF.A.1a, F-BF.A.2, F-LE.A.2), with a free PDF to print.

Key takeaway: This worksheet helps Algebra 1 students practice finding terms and writing rules for arithmetic and geometric sequences.

Topic: Arithmetic and Geometric Sequences (Algebra 1)
Content: 21 worksheet problems, from easy to challenging
Standards: F-IF.A.3, F-BF.A.1a, F-BF.A.2, F-LE.A.2
Answers: On a separate page, so you can try every worksheet problem first
Format: Free printable PDF, US Letter, black and white

This worksheet helps Algebra 1 students practice finding terms and writing rules for arithmetic and geometric sequences. Also, the problems begin with continuing a pattern and end with real situations.

There are 21 worksheet problems. Worksheet problems 1 to 4 find the next terms, classify a sequence, and find a difference and a factor. Also, worksheet problems 5 to 9 write explicit and recursive rules. Worksheet problems 10 to 13 find a term number, use two given terms, and compute a term from a recursive rule. Also, worksheet problems 14 and 15 use a graph and a pattern of squares. Worksheet problems 16 to 21 apply sequences to savings, bounce heights, bacteria, stadium seats, a raise, and two pay plans. Allow about 45 minutes.

Use these 20 problems to master arithmetic and geometric sequences (Common Core F-IF.A.3, F-BF.A.1a, F-BF.A.2, and F-LE.A.2). You will find common differences and ratios, write recursive and explicit rules, read sequences from graphs and patterns, and model growth in real situations. They move from quick warm-ups to multistep word problems, so slow down on the last six and check that each answer makes sense.

Algebra 1 · Common Core State Standards: F-IF.A.3, F-BF.A.1a, F-BF.A.2, F-LE.A.2

Arithmetic and geometric sequences worksheet: worksheet problems 1 to 10

1 – Next terms

A pattern.

First, find the next three terms of the sequence $8, 13, 18, 23$.

2 – Arithmetic or geometric?

Next, decide whether $3, 6, 12, 24$ is arithmetic, geometric, or neither.

3 – Common difference

Then, find the common difference of $40, 33, 26, 19$.

4 – Common ratio

Population at the start of each decade.

Now, a town’s population multiplies by the same factor each decade: $2500, 3000, 3600, 4320$. Find the common ratio and the population at the start of the next decade.

5 – Explicit rule

Also, write an explicit rule for $7, 12, 17, 22$.

6 – A distant term

First, find the 20th term of the sequence $4, 7, 10, 13$.

7 – Recursive rule

Next, write a recursive rule for $10, 6, 2, -2$.

8 – Geometric explicit rule

Terms of a geometric sequence.

Then, write an explicit rule for $3, 6, 12, 24$ and find the 6th term.

9 – Fractional ratio

Now, write an explicit rule for $81, 27, 9, 3$ and find the 5th term.

10 – Which term?

Also, which term of $5, 10, 15, 20$ equals 95?

Worksheet problems 11 and up

11 – Two terms given

First, an arithmetic sequence has $a_{2} = 5$ and $a_{6} = 21$. Find the common difference and a rule for $a_{n}$.

12 – Evaluate a recursive rule

Next, a sequence is defined by $a_{1} = 2$ and $a_{n} = 3a_{n-1} – 1$. Find $a_{4}$.

13 – Is 100 in the sequence?

Then, is 100 a term of $7, 12, 17, 22$?

14 – Graph of a sequence

Four terms of a sequence.

Now, the graph shows the first four terms of an arithmetic sequence. Write its explicit rule.

15 – Pattern of squares

Three figures.

Also, the figures use 5, 7, and 9 squares. Write a rule for the number of squares in Figure $n$ and find Figure 10.

16 – Weekly savings

Maya saves $15 in week 1, $20 in week 2, $25 in week 3, and keeps adding $5 more each week. Write a rule for the amount in week $n$. Find week 12 and the week in which she saves $100.

17 – Bouncing ball

Bounce heights in feet.

Next, a ball’s bounce heights are 64 ft, 32 ft, 16 ft, 8 ft. Write an explicit rule and find the 5th bounce.

Worksheet problem 18 and after

18 – Bacteria growth

Then, a lab starts with 250 bacteria, and the number doubles every hour. Let $a_{1} = 250$ be the count at hour 0. Write an explicit rule and find the count after 5 hours.

19 – Stadium seats

Row 1 of a stadium section has 20 seats and each row has 2 more seats than the row before. Write a rule for row $n$. How many seats are in row 25? Which row has 100 seats?

20 – Yearly raise

Liam’s job pays $40,000 in year 1 and gives a 5% raise each year. Write a rule for his pay in year $n$ and find his pay in year 4.

21 – Two pay plans

Daily pay for two plans.

Plan A pays $2 on day 1 and $3 more each day. Plan B pays $1 on day 1 and doubles each day. Write rules for both plans and find the first day when Plan B pays more than Plan A.

Key facts behind this arithmetic and geometric sequences worksheet

Key facts

A sequence is a function whose inputs are the positions 1, 2, 3, and so on.
Arithmetic: add d each time; aₙ = a₁ + (n – 1)d.
Geometric: multiply by r each time; aₙ = a₁ · r^(n-1).
Then, a recursive rule gives each term from the previous term; an explicit rule gives any term directly.
Arithmetic sequences are linear; geometric sequences are exponential.

Common mistakes with this arithmetic and geometric sequences worksheet

Students multiply the common difference by n to find a distant term. However, only n minus 1 differences are added to the first term, so count the steps carefully.
Students treat a percent raise as a fixed amount added each year. Instead, a percent raise multiplies by a factor each year, so the sequence is geometric.
Students apply a recursive rule once and stop. First build each term in order, and then write every term before you go on to the next.

Tips before you start the worksheet problems

In worksheet problem 21, make a table of both plans for the first six days. As a result, the comparison shows when doubling overtakes adding.
For a pattern of figures, write the number of squares for the first three figures, then look for the constant change.
Teachers can ask students to write both a recursive and an explicit rule for the same sequence and explain when each is useful.

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Arithmetic and geometric sequences worksheet: frequently asked questions

How do you write a rule for a growing pattern of tiles?

Count the tiles in each figure, find how many are added each time, and write an arithmetic rule. For figures with 4, 7, and 10 tiles, the difference is 3, so the rule is $a_{n} = 4 + 3(n – 1)$, or $3n + 1$.

How do you model a yearly percent raise as a sequence?

Multiply the starting pay by one plus the rate for each year after the first. In other words, a salary of 30,000 dollars with a 3 percent raise gives $a_{n} = 30000(1.03)^{n-1}$. The ratio is 1.03, so the sequence is geometric.

How do you tell whether a number is in a sequence?

Solve the explicit rule for n, and see whether the result is a positive whole number. For the rule $a_{n} = 4n + 1$, setting it to 50 gives n = 12.25, so 50 is not a term. Setting it to 49 gives n = 12.

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