Algebra 1

Sequences & Patterns

Numbers in a pattern. That is what a sequence is — a list of numbers that follow a rule. Once you see the rule, you can predict the 10th term, the 100th term, or even the millionth term without writing them all out.

Let’s explore the two most important types of sequences in algebra.

Part 1: Arithmetic Sequences — Adding the Same Amount

An arithmetic sequence is a list of numbers where you add the same value to get from one term to the next. That value is called the common difference, d.

For example: 3, 7, 11, 15, 19, … has a common difference of d = 4.

The formula for the nth term is:

a_n = a_1 + (n - 1) \cdot d

where a_1 is the first term and d is the common difference.

First term (a_1)2

-1010

Common difference (d)3

-55

a_n = 2 + (n - 1) \cdot 3

Try This

Notice these things as you drag the sliders:

The points always form a straight line — arithmetic sequences are linear!
d > 0: The sequence goes up (increasing)
d < 0: The sequence goes down (decreasing)
d = 0: Every term is the same — a constant sequence
Changing a_1 slides the whole line up or down

Part 2: Geometric Sequences — Multiplying by the Same Amount

A geometric sequence is a list of numbers where you multiply by the same value to get from one term to the next. That value is called the common ratio, r.

For example: 2, 6, 18, 54, 162, … has a common ratio of r = 3.

The formula for the nth term is:

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

First term (a_1)2

0.510

Common ratio (r)2

0.53

a_n = 2 \cdot 2^{(n-1)}

Try This

Play with the ratio and notice:

r > 1: The sequence grows faster and faster — exponential growth!
r = 1: Every term is the same (multiplying by 1 changes nothing)
0.5 < r < 1: The sequence shrinks toward zero — exponential decay
The larger the ratio, the more dramatic the curve

Part 3: Side by Side — Arithmetic vs. Geometric

This is where it gets interesting. Let’s put both sequences on the same graph so you can directly compare how they grow.

Starting value (both)2

110

Common difference (d)3

Common ratio (r)1.5

1.12.5

\text{Arithmetic: } a_n = 2 + (n-1) \cdot 3

\text{Geometric: } a_n = 2 \cdot 1.5^{(n-1)}

Connection

The key difference:

The arithmetic sequence forms a straight line — it grows at a constant rate
The geometric sequence forms a curve — it grows at an accelerating rate
Both start at the same place, but the geometric sequence eventually pulls away dramatically

This is the exact same idea as linear vs. exponential growth, just viewed through the lens of sequences instead of functions!

Part 4: Decaying Geometric Sequences

When the common ratio is between 0 and 1, something different happens — the geometric sequence shrinks instead of growing. This is called exponential decay.

Starting value (a_1)50

10100

Decay ratio (r)0.8

0.50.95

a_n = 50 \cdot 0.8^{(n-1)}

Connection

Real-world decay examples:

A bouncing ball that reaches 80% of its previous height each bounce (r = 0.8)
Medicine leaving your body — half-life means r = 0.5 each period
A car losing 15% of its value each year (r = 0.85)

Notice how the sequence gets closer and closer to zero but never quite reaches it? That is a hallmark of exponential decay.

Part 5: Finding the Pattern

Given a sequence, how do you figure out if it is arithmetic or geometric?

Arithmetic test: Subtract consecutive terms. If the difference is always the same, it is arithmetic.

5, 8, 11, 14, 17 … differences: 3, 3, 3, 3 — Arithmetic! (d = 3)

Geometric test: Divide consecutive terms. If the ratio is always the same, it is geometric.

3, 6, 12, 24, 48 … ratios: 2, 2, 2, 2 — Geometric! (r = 2)

Challenge

Classify each sequence as arithmetic or geometric, then find the 10th term:

4, 10, 16, 22, 28, …
5, 15, 45, 135, …
100, 90, 80, 70, …
1000, 500, 250, 125, …

Hints:

For #1: What is the common difference? Use a_n = a_1 + (n-1)d with n = 10
For #2: What is the common ratio? Use a_n = a_1 * r^(n-1) with n = 10
For #3: Subtracting gives the same result each time
For #4: Dividing gives the same result each time

Part 6: The Formulas at a Glance

Let’s compare the two formulas with all parameters adjustable.

First term (a_1)3

110

Difference (d)2

Ratio (r)1.5

1.22

\text{Arithmetic: } a_n = 3 + (n-1) \times 2

\text{Geometric: } a_n = 3 \times 1.5^{(n-1)}

The dashed horizontal line shows the shared starting value. Both sequences begin at the same point, but their paths diverge based on whether they add or multiply at each step.

Wrapping Up

Property	Arithmetic Sequence	Geometric Sequence
Rule	Add the same value (d)	Multiply by the same value (r)
Formula	a_n = a_1 + (n-1)d	a_n = a_1 * r^(n-1)
Graph shape	Straight line	Exponential curve
Growth	Constant (linear)	Accelerating (exponential)
Example	2, 5, 8, 11, 14, …	2, 6, 18, 54, 162, …

Challenge

Final Challenge:

Write the first 6 terms of an arithmetic sequence with a_1 = 7 and d = -3. What is the formula for the nth term?
Write the first 6 terms of a geometric sequence with a_1 = 4 and r = 2. What is the 8th term?
A sequence starts 100, 80, 64, 51.2, … Is it arithmetic or geometric? What is the common ratio or difference? Predict the next two terms.
You save $20 the first week, then $5 more each week after that (arithmetic). Your friend saves $5 the first week but doubles their savings each week (geometric). After 8 weeks, who has saved more in that single week? Use the sliders above to check!

Sequences are everywhere — music patterns, savings plans, population models, even the way a virus spreads. Once you learn to spot the pattern, you can predict what comes next.

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