Algebra 1 Arithmetic and Geometric Sequences Worksheet Solutions (PDF)

Algebra 1 math worksheet solutions: Arithmetic and Geometric Sequences – Maths PDF

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This sequences practice solutions 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.

Sequences practice solutions (Algebra 1) – first page of the printable PDF

This answer key explains every problem on the sequences worksheet. Final answers are in bold, and the lines above them show how each result was found. Use it to correct mistakes and to look for patterns in the errors you make most often.

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

Sequences practice solutions: exercises 1 to 10

Exercise 1 – Next terms

The common difference is $13 – 8 = 5$.

Next terms: $23 + 5 = 28$, $28 + 5 = 33$, $33 + 5 = 38$.

28, 33, 38

Exercise 2 – Arithmetic or geometric?

Differences: 3, 6, 12 (not constant). Ratios: $\frac{6}{3} = 2$, $\frac{12}{6} = 2$, $\frac{24}{12} = 2$ (constant).

Geometric, ratio 2

Exercise 3 – Common difference

$33 – 40 = -7$, $26 – 33 = -7$, $19 – 26 = -7$.

$d = -7$

Exercise 4 – Common ratio

$\frac{3000}{2500} = 1.2$, $\frac{3600}{3000} = 1.2$, $\frac{4320}{3600} = 1.2$.

Next decade: $4320 \cdot 1.2 = 5184$.

$r = 1.2$; 5,184 people

Exercise 5 – Explicit rule

Common difference 5.

The common difference is 5 and $a_{1} = 7$.

$a_{n} = 7 + (n – 1)(5) = 7 + 5n – 5 = 5n + 2$.

$a_{n} = 5n + 2$

Exercise 6 – A distant term

The rule is $a_{n} = 4 + (n – 1)(3) = 3n + 1$.

$a_{20} = 3(20) + 1 = 61$.

Exercise 7 – Recursive rule

The common difference is $6 – 10 = -4$.

$a_{1} = 10$, $a_{n} = a_{n-1} – 4$

Exercise 8 – Geometric explicit rule

The ratio is 2 and the first term is 3: $a_{n} = 3 \cdot 2^{n-1}$.

$a_{6} = 3 \cdot 2^{5} = 3 \cdot 32 = 96$.

$a_{n} = 3 \cdot 2^{n-1}$; 96

Exercise 9 – Fractional ratio

The ratio is $\frac{27}{81} = \frac{1}{3}$.

$a_{n} = 81\left(\frac{1}{3}\right)^{n-1}$.

$a_{5} = 81 \cdot \frac{1}{81} = 1$.

$a_{n} = 81\left(\frac{1}{3}\right)^{n-1}$; 1

Exercise 10 – Which term?

The rule is $a_{n} = 5n$. Solve $5n = 95$: $n = 19$.

The 19th term

Exercises 11 and up

Exercise 11 – Two terms given

From term 2 to term 6 is 4 steps: $d = \frac{21 – 5}{4} = 4$.

Then $a_{1} = 5 – 4 = 1$, so $a_{n} = 1 + (n – 1)(4) = 4n – 3$.

$d = 4$; $a_{n} = 4n – 3$

Exercise 12 – Evaluate a recursive rule

Applying the rule three times.

$a_{2} = 3(2) – 1 = 5$, $a_{3} = 3(5) – 1 = 14$, $a_{4} = 3(14) – 1 = 41$.

Exercise 13 – Is 100 in the sequence?

The rule is $a_{n} = 5n + 2$. Solve $5n + 2 = 100$: $5n = 98$, so $n = 19.6$.

The position must be a whole number.

Exercise 14 – Graph of a sequence

The terms are 5, 8, 11, 14, so $d = 3$ and $a_{1} = 5$.

$a_{n} = 5 + (n – 1)(3) = 3n + 2$.

$a_{n} = 3n + 2$

Exercise 15 – Pattern of squares

The differences are 2, so $a_{n} = 5 + (n – 1)(2) = 2n + 3$.

$a_{10} = 2(10) + 3 = 23$.

$a_{n} = 2n + 3$; 23 squares

Exercise 16 – Weekly savings

Weeks 1 to 4.

$a_{n} = 15 + (n – 1)(5) = 5n + 10$.

Week 12: $5(12) + 10 = 70$. Solve $5n + 10 = 100$: $n = 18$.

$a_{n} = 5n + 10$; $70; week 18

Exercise 17 – Bouncing ball

The ratio is $\frac{1}{2}$.

$a_{n} = 64\left(\frac{1}{2}\right)^{n-1}$. $a_{5} = 64 \cdot \frac{1}{16} = 4$.

$a_{n} = 64\left(\frac{1}{2}\right)^{n-1}$; 4 ft

Exercise 18 – Bacteria growth

$a_{n} = 250 \cdot 2^{n-1}$. After 5 hours is term 6.

$a_{6} = 250 \cdot 2^{5} = 250 \cdot 32 = 8000$.

$a_{n} = 250 \cdot 2^{n-1}$; 8,000 bacteria

Exercise 19 – Stadium seats

$a_{n} = 20 + (n – 1)(2) = 2n + 18$.

Row 25: $2(25) + 18 = 68$. Solve $2n + 18 = 100$: $n = 41$.

$a_{n} = 2n + 18$; 68 seats; row 41

Exercise 20 – Yearly raise

A 5% raise multiplies by 1.05: $a_{n} = 40000(1.05)^{n-1}$.

$a_{4} = 40000(1.05)^{3} = 40000(1.157625) = 46305$.

$a_{n} = 40000(1.05)^{n-1}$; $46,305

Exercise 21 – Two pay plans

Plan A is arithmetic: $a_{n} = 3n – 1$. Plan B is geometric: $b_{n} = 2^{n-1}$.

Compare: day 4 is $11 versus $8, day 5 is $14 versus $16.

$a_{n} = 3n – 1$; $b_{n} = 2^{n-1}$; day 5

Key facts behind this sequences practice solutions

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).
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.

How to use this sequences practice solutions

Cover each solution, solve the exercise again, then uncover it and compare every line. If your answer differs, find the first step that changed and redo the exercise from there.

Download the sequences practice solutions as a PDF with the button below and keep it with your notes.

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Algebra 1 Arithmetic and Geometric Sequences Worksheet Solutions (PDF)

Sequences practice solutions: exercises 1 to 10

Exercises 11 and up

Key facts behind this sequences practice solutions

How to use this sequences practice solutions

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