Algebra I: Arithmetic and Geometric Sequences

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Tell arithmetic sequences (a common difference) from geometric ones (a common ratio), list terms from a recursive rule in aₙ or f(n) notation, write the nth-term formula for each kind, switch between the recursive rule and the formula, and find missing terms or which term a number is. Aligned to HSF-IF.A.3, HSF-BF.A.2 and HSF-LE.A.2.

What you’ll learn

Arithmetic sequences add a common difference $d$ ; geometric sequences multiply by a common ratio $r$ .
A recursive rule gives the first term and how to get each term from the one before: $a_n=a_{n-1}+d$ or $a_n=r\cdot a_{n-1}$ .
Arithmetic $n$ th term: $a_n=a_1+(n-1)d$ . Geometric $n$ th term: $a_n=a_1\cdot r^{n-1}$ .
Use $n-1$ , the number of steps from the first term, not $n$ .
From two arithmetic terms, $d$ is the change in the terms divided by the number of steps between them.
Arithmetic sequences are linear and geometric sequences exponential, with term numbers $n=1, 2, 3, \ldots$

Aligned to HSF-IF.A.3, HSF-BF.A.2, HSF-LE.A.2

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Algebra I: Arithmetic and Geometric Sequences

Free

Grades 8–10AlgebraGuided journey: Discover, Learn, Practice, Apply

What you’ll learn

Arithmetic sequences add a common difference $d$ ; geometric sequences multiply by a common ratio $r$ .
A recursive rule gives the first term and how to get each term from the one before: $a_n=a_{n-1}+d$ or $a_n=r\cdot a_{n-1}$ .
Arithmetic $n$ th term: $a_n=a_1+(n-1)d$ . Geometric $n$ th term: $a_n=a_1\cdot r^{n-1}$ .
Use $n-1$ , the number of steps from the first term, not $n$ .
From two arithmetic terms, $d$ is the change in the terms divided by the number of steps between them.
Arithmetic sequences are linear and geometric sequences exponential, with term numbers $n=1, 2, 3, \ldots$

Aligned to HSF-IF.A.3, HSF-BF.A.2, HSF-LE.A.2

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Share to Google Classroom

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Reference

Algebra I: Arithmetic and Geometric Sequences

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Think about this

Why does a recursive rule need a starting term?

Each term adds 4 to the one before: an arithmetic sequence with common difference d = 4.

Learn

Each card starts with a real-life picture of the idea, works one example step by step, then shows a mistake to avoid and a tool to try it with. Try the steps yourself before moving to Practice.

Arithmetic or geometric?

Cups and paper

10

Worked example: is $3, 7, 11, 15, 19, \ldots$ arithmetic, geometric or neither?

7-3=4\text{,}

Check the ratios too: $\tfrac{7}{3}$ and $\tfrac{11}{7}$ differ, so it is not geometric.

Each term adds 4 to the one before: an arithmetic sequence with common difference d = 4.

5, 15, 45, 135, \ldots

1, 3, 6, 10, \ldots

Big mistake to avoid

20, 17, 14

Cool math secret

Subtract to test for arithmetic, divide to test for geometric. Arithmetic sequences plot as points on a line; geometric ones plot on an exponential curve.

Worked example

5, 9, 13, 17, \ldots

arithmetic, geometric or neither?

Subtract each term from the next.
$9 - 5 = 13 - 9 = 17 - 13 = 4$
Name it with its common difference.
$\text{arithmetic, } d = 4$

Answer

\text{arithmetic, } d = 4

Now you try

Now classify

80, 40, 20, 10, \ldots

Arithmetic,

d = -40

Geometric,

r = \tfrac{1}{2}

Geometric,

r = 2

Neither

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• Arithmetic sequence formula : the common difference at work. • Geometric sequence formula : the common ratio at work.

Terms from a recursive rule

The chain

a_{n-1}

Worked example: list the first four terms of $a_1=5\text{,}$ $a_n=a_{n-1}+3\text{.}$

a_1=5\text{.}

The terms are $5, 8, 11, 14\text{:}$ arithmetic with $d=3\text{.}$

The rule a₁ = 5, aₙ = aₙ₋₁ + 3 gives the start and the step. Each term is built from the one before it.

a_1=2\text{,}

f(1)=40\text{,}

Big mistake to avoid

a_{n-1}

Cool math secret

a_1=5\text{,}

Worked example

List the first four terms of

a_1 = 4\text{,}

a_n = a_{n-1} - 3\text{.}

The first term is given.
$a_1 = 4$
Each next term is the one before, minus $3\text{.}$
$a_2 = 4 - 3 = 1, \quad a_3 = 1 - 3 = -2$
One more step.
$a_4 = -2 - 3 = -5$

Answer

4,\ 1,\ -2,\ -5

Now you try

Now:

f(1) = 3

and

f(n) = 2 \cdot f(n-1)\text{.}

What is

f(4)\text{?}

12

24

48

9

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x

The nth term of an arithmetic sequence

The theater rows

20

The nth term of an arithmetic sequence

a_n=a_1+(n-1)d

Worked example: write the $n$ th term of $7, 11, 15, 19, \ldots$ and find $a_{20}\text{.}$

a_1=7

Check: $n=1$ gives $7$ and $n=2$ gives $11\text{,}$ the first two terms.

Plotted as (n, aₙ), the terms lie on a line with slope d = 4: aₙ = 4n + 3. Only whole-number n are terms.

50, 44, 38, 32, \ldots

a_n=4n+3

Big mistake to avoid

n-1\text{,}

Cool math secret

n

Worked example

Write the

n

th term of

7, 11, 15, 19, \ldots

and find

a_{20}\text{.}

Find the first term and the common difference.
$a_1 = 7, \quad d = 4$
Use $a_n = a_1 + (n - 1)d\text{.}$
$a_n = 7 + (n - 1)4 = 4n + 3$
Substitute $n = 20\text{.}$
$a_{20} = 4(20) + 3 = 83$

Answer

a_n = 4n + 3, \quad a_{20} = 83

Now you try

Now write the

n

th term of

50, 44, 38, 32, \ldots

a_n = 56 - 6n

a_n = 50 - 6n

a_n = 6n + 44

a_n = 50(-6)^{n-1}

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a_n=a_1+(n-1)d

The nth term of a geometric sequence

The folded paper

0.1

The nth term of a geometric sequence

a_n=a_1\cdot r^{\,n-1}

Worked example: write the $n$ th term of $3, 6, 12, 24, \ldots$ and find $a_8\text{.}$

a_1=3

Check: $n=1$ gives $3(2)^0=3\text{,}$ the first term.

Plotted as (n, aₙ), each term doubles the last: aₙ = 3(2)ⁿ⁻¹. The points sit on an exponential curve.

162, 54, 18, 6, \ldots

a_n=3(2)^{n-1}

Big mistake to avoid

n-1\text{,}

Cool math secret

a_1

Worked example

Write the

n

th term of

3, 6, 12, 24, \ldots

and find

a_8\text{.}

Find the first term and the common ratio.
$a_1 = 3, \quad r = 2$
Use $a_n = a_1 \cdot r^{n-1}\text{.}$
$a_n = 3(2)^{n-1}$
Substitute $n = 8\text{.}$
$a_8 = 3(2)^7 = 3(128) = 384$

Answer

a_n = 3(2)^{n-1}, \quad a_8 = 384

Now you try

Now write the

n

th term of

162, 54, 18, 6, \ldots

a_n = 162\left(\tfrac{1}{3}\right)^{n-1}

a_n = 162(3)^{n-1}

a_n = 162\left(\tfrac{1}{3}\right)^{n}

a_n = 162 - 108(n - 1)

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a_n=a_1r^{n-1}

Switch between the two rules

Two ways to the same place

n

Worked example: write a recursive rule for $a_n=5n-2\text{.}$

a_1=5(1)-2=3\text{.}

Both rules give $3, 8, 13, 18, \ldots$

Both give 3, 8, 13, 18, … The recursive rule walks from term to term; the nth-term formula finds a₁₀₀ in one step.

a_1=10\text{,}

a_1=4\text{,}

Big mistake to avoid

a_1\text{.}

Cool math secret

a_{100}=5(100)-2=498

Worked example

Write a recursive rule for

a_n = 5n - 2\text{.}

Find the first term.
$a_1 = 5(1) - 2 = 3$
The number on $n$ is the common difference.
$d = 5$
Start, then step.
$a_1 = 3, \quad a_n = a_{n-1} + 5$

Answer

a_1 = 3, \quad a_n = a_{n-1} + 5

Now you try

Now write the

n

th-term formula for

a_1 = 4\text{,}

a_n = 3 \cdot a_{n-1}\text{.}

a_n = 3(4)^{n-1}

a_n = 4(3)^{n-1}

a_n = 4 + 3(n - 1)

a_n = 4(3)^n

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5n-2\text{.}

Missing terms and which term

The mile markers

3

Worked example: an arithmetic sequence has $a_3=14$ and $a_7=30\text{.}$ Find $a_n\text{,}$ then find which term is $50\text{.}$

3

The difference is the slope between the two points $(3, 14)$ and $(7, 30)\text{.}$

From a₃ = 14 to a₇ = 30 is 4 steps and 16 up, so d = 16 ÷ 4 = 4. Two steps back from a₃ gives a₁ = 6.

43

a_1=5

Big mistake to avoid

a_3

Cool math secret

d

Worked example

An arithmetic sequence has

a_3 = 14

and

a_7 = 30\text{.}

Find

a_n\text{,}

then find which term is

50\text{.}

From term $3$ to term $7$ is $4$ steps.
$d = \frac{30 - 14}{7 - 3} = \frac{16}{4} = 4$
Walk back two steps from $a_3\text{.}$
$a_1 = 14 - 2(4) = 6$
Write $a_n\text{.}$
$a_n = 6 + 4(n - 1) = 4n + 2$
Solve $a_n = 50\text{.}$
$4n + 2 = 50 \quad\Rightarrow\quad n = 12$

Answer

a_n = 4n + 2, \quad a_{12} = 50

Now you try

Now: an arithmetic sequence has

a_2 = 9

and

a_6 = 25\text{.}

Which term is

61\text{?}

a_{15}

a_{16}

a_{14}

a_{61/4}

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4n+2\text{.}

Key points to remember

Arithmetic sequences add a common difference $d\text{;}$ geometric sequences multiply by a common ratio $r\text{.}$
A recursive rule gives the first term and how to get each term from the one before: $a_n=a_{n-1}+d$ or $a_n=r\cdot a_{n-1}\text{.}$
Arithmetic $n$ th term: $a_n=a_1+(n-1)d\text{.}$ Geometric $n$ th term: $a_n=a_1\cdot r^{n-1}\text{.}$
Use $n-1\text{,}$ the number of steps from the first term, not $n\text{.}$
From two arithmetic terms, $d$ is the change in the terms divided by the number of steps between them.
Arithmetic sequences are linear and geometric sequences exponential, with term numbers $n=1, 2, 3, \ldots$

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Algebra I: Arithmetic and Geometric Sequences

What you’ll learn

Algebra I: Arithmetic and Geometric Sequences

What you’ll learn

Where you are

Units

Get the free 7-Day Study Reset

Where you are

Units

Get the free 7-Day Study Reset

Learning acts

Arithmetic or geometric?

Terms from a recursive rule

The nth term of an arithmetic sequence

The nth term of a geometric sequence

Switch between the two rules

Missing terms and which term

Key points to remember

Practice what you learned

Practice for you

Flashcards

Apply what you learned

Explain it in your own words