Identify and extend numeric and algebraic patterns - TACHS Math

SECTION 1

Historical Context & Motivation

People have been fascinated by patterns for thousands of years. Ancient builders used repeating designs on walls. Early farmers noticed patterns in the seasons. Mathematicians noticed patterns in numbers, too, and began writing rules to describe them.

Understanding patterns lets you predict what comes next without counting every single item. That skill saves time and helps you solve real problems, from planning a budget to cracking a test question.

~300 BC • Euclid's Elements

The Greek mathematician Euclid wrote about number patterns, including even and odd numbers, in one of the earliest math textbooks ever created.

~600 AD • Indian Mathematicians

Scholars in India studied arithmetic sequences (patterns that grow by the same amount) and used them to solve everyday problems in trade and astronomy.

1202 • Fibonacci's Liber Abaci

Leonardo of Pisa (Fibonacci) introduced his famous sequence — 1, 1, 2, 3, 5, 8, … — where each number is the sum of the two before it. This pattern appears in sunflowers, pinecones, and seashells.

1700s • Carl Friedrich Gauss

As a young student, Gauss amazed his teacher by quickly adding the numbers 1 through 100. He spotted a pattern that let him find the sum in seconds instead of minutes.

The big question these thinkers were all asking is the same question you will explore in this lesson: When you see a list of numbers, how do you figure out the rule and predict what comes next?

SECTION 2

Core Principles & Definitions

Before you start solving pattern problems, you need to know a few key ideas. These are the building blocks for everything else in this lesson.

Sequence

A sequence is an ordered list of numbers that follow a rule. Each number in the list is called a term.

Common Difference

The common difference is the amount you add (or subtract) each time to go from one term to the next. If this amount stays the same, you have an arithmetic sequence.

Common Ratio

The common ratio is the number you multiply by each time. If this multiplier stays the same, you have a geometric sequence.

Pattern Rule (Expression)

A pattern rule is an algebraic expression that lets you find any term using its position number, often called n.

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KEY TAKEAWAY

Think of a pattern like a song playlist on repeat. Once you know the rule (the playlist order), you can predict any song without listening to every one before it. A pattern rule is your shortcut to jump straight to any term in the sequence.

SECTION 3

Visual Explanation — Seeing the Pattern

The best way to understand a pattern is to see it. The diagram below shows an arithmetic sequence — the kind where you add the same number each time.

Each box shows a term, and the arrows show the common difference of +4 between consecutive terms. Subtract any term from the one after it to find this difference.

Notice that the difference between every pair of neighbors is exactly 4. That steady jump is what makes this an arithmetic sequence. If the differences changed each time, you would need a different type of rule.

SECTION 4

Mathematical Framework

Once you spot the pattern type, you can write a formula (a math rule) to find any term without listing them all. Here are the two main formulas you need.

ARITHMETIC SEQUENCE RULE

a(n) = a(1) + (n − 1) × d

a(n) = the term you want to find, a(1) = the first term, n = the position number, d = the common difference.

Use this formula when the sequence grows (or shrinks) by adding the same amount each time. For example, 3, 7, 11, 15, … has a(1) = 3 and d = 4.

GEOMETRIC SEQUENCE RULE

a(n) = a(1) × r⁽ⁿ⁻¹⁾

a(n) = the term you want, a(1) = the first term, r = the common ratio (the number you multiply by), n = the position number.

Use this formula when the sequence grows (or shrinks) by multiplying by the same amount each time. For example, 2, 6, 18, 54, … has a(1) = 2 and r = 3.

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Quick Tip

Not sure which formula to use? Try subtracting first. If the differences are all the same, it's arithmetic. If not, try dividing each term by the one before it. If the quotients are all the same, it's geometric.

SECTION 5

Types of Patterns You'll See on the TACHS

TACHS pattern questions come in several flavors. Knowing the different types helps you pick the right strategy quickly. The diagram below compares the two most common types side by side.

The arithmetic sequence (top) has evenly spaced dots because you add the same amount each time. The geometric sequence (bottom) has dots that spread farther apart because multiplying makes numbers grow much faster.

Pattern Type	How It Works	Example
Arithmetic	Add (or subtract) the same number each time	4, 9, 14, 19, 24, … (d = +5)
Geometric	Multiply (or divide) by the same number each time	3, 12, 48, 192, … (r = ×4)
Repeating	A group of numbers cycles over and over	1, 4, 7, 1, 4, 7, 1, 4, 7, …
Other / Two-Step	Uses a combination of operations (add then multiply, etc.)	2, 5, 11, 23, … (×2 + 1)

Common pattern types on the TACHS

SECTION 6

Worked Example

Let's walk through a complete example, just like you'd see on the TACHS. Pay attention to each step so you can repeat the process on your own.

Find the 10th term of the sequence: 6, 11, 16, 21, 26, …

1
Step 1 — Identify the Pattern Type

Subtract each term from the next: 11 − 6 = 5, 16 − 11 = 5, 21 − 16 = 5. The difference is always 5, so this is an arithmetic sequence.

d = 5
2
Step 2 — Write Down What You Know

First term: a(1) = 6. Common difference: d = 5. We want the 10th term, so n = 10.

a(1) = 6, d = 5, n = 10
3
Step 3 — Plug into the Formula

Use the arithmetic sequence formula: a(n) = a(1) + (n − 1) × d. Substitute the values: a(10) = 6 + (10 − 1) × 5.

a(10) = 6 + (9) × 5
4
Step 4 — Simplify

First, multiply: 9 × 5 = 45. Then add: 6 + 45 = 51.

a(10) = 51
5
Step 5 — Check Your Answer

You can verify by listing terms: 6, 11, 16, 21, 26, 31, 36, 41, 46, 51. The 10th term is indeed 51. ✓

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REMEMBER

Always check your answer by counting a few terms by hand. On a multiple-choice test, this quick check can save you from picking the wrong answer.

SECTION 7

Strategies, Strengths & Pitfalls

Different strategies work better for different pattern types. Knowing when to use each one will help you work faster and avoid common mistakes.

Strategy	When to Use It	Watch Out For
Subtract consecutive terms	When you think the pattern adds the same number each time (arithmetic)	Don't assume it's arithmetic until you check at least 2 or 3 differences
Divide consecutive terms	When differences are not equal — the pattern might multiply (geometric)	If the ratio doesn't stay the same, try a two-step rule or look for a repeating cycle
Use the formula	When you need a term far down the line (like the 50th term)	Remember to use (n − 1), not just n, in the formula
Make a table	When the pattern seems tricky or uses more than one operation	Double-check each row before moving on

Strategy comparison for pattern problems

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KEY TAKEAWAY

Think of pattern strategies like tools in a toolbox. A hammer (subtraction) is great for nails (arithmetic patterns), but you'd use a wrench (division) for bolts (geometric patterns). Pick the right tool before you start working.

SECTION 8

Connection to Advanced Topics

The pattern skills you are learning now are the same ones used in high school algebra and beyond. When you study linear functions in algebra, you will see that an arithmetic sequence is really just a straight line in disguise. And geometric sequences lead to exponential functions, which model things like population growth and compound interest.

What You Learn Now	Where It Goes Next
Arithmetic sequences (add a constant)	Linear functions: y = mx + b, slope, graphing lines
Geometric sequences (multiply by a constant)	Exponential functions: y = a × rˣ, compound interest, growth/decay
Writing pattern rules with n	Writing and evaluating algebraic functions, input-output tables
Finding the common difference d	Slope of a line (rise over run), rate of change

How pattern skills build toward high school algebra

So by mastering patterns now, you're actually getting a head start on some of the biggest topics in algebra. The common difference d will become the slope of a line, and the first term a(1) will connect to the y-intercept. Pretty cool, right?

SECTION 9

Practice Problems

Try these five problems on your own. They start easy and get harder. After each one, check the answer to see if you're on the right track.

Question 1: CONCEPTUAL

Look at this sequence: 8, 14, 20, 26, 32, … Is this an arithmetic sequence, a geometric sequence, or neither? How can you tell?

Show answer

This is an arithmetic sequence. You can tell by subtracting each term from the next: 14 − 8 = 6, 20 − 14 = 6, 26 − 20 = 6, 32 − 26 = 6. Because the difference is always 6, the sequence adds the same amount each time. That makes the common difference d = 6.

Question 2: BASIC CALCULATION

What is the next term in the sequence 4, 12, 36, 108, …?

Show answer

Divide each term by the one before it: 12 ÷ 4 = 3, 36 ÷ 12 = 3, 108 ÷ 36 = 3. The common ratio is 3, so this is a geometric sequence. To find the next term, multiply the last given term by 3: 108 × 3 = 324. The next term is 324.

Question 3: INTERMEDIATE

Find the 15th term of the arithmetic sequence: 7, 12, 17, 22, 27, …

Show answer

First, identify the common difference: 12 − 7 = 5, so d = 5. The first term is a(1) = 7, and you need n = 15. Use the formula a(n) = a(1) + (n − 1) × d. Substitute: a(15) = 7 + (15 − 1) × 5 = 7 + 14 × 5 = 7 + 70 = 77. The 15th term is 77.

Question 4: APPLIED

Maria saves money each week. In Week 1 she saves $10, in Week 2 she saves $15, in Week 3 she saves $20, and so on. How much will she save in Week 12?

Show answer

Maria's savings form an arithmetic sequence: 10, 15, 20, 25, … The common difference is d = 5 (she saves $5 more each week). Use the formula: a(12) = 10 + (12 − 1) × 5 = 10 + 11 × 5 = 10 + 55 = 65. Maria will save $65 in Week 12. Notice how the formula lets you jump straight to Week 12 without calculating every week in between.

Question 5: CRITICAL THINKING

A sequence starts 2, 5, 11, 23, 47, … This is not arithmetic (the differences change) and not geometric (the ratios change). Can you figure out the rule and predict the 6th term?

Show answer

Look at how each term relates to the one before it. Try multiplying by 2 and then adding 1: 2 × 2 + 1 = 5 ✓, 5 × 2 + 1 = 11 ✓, 11 × 2 + 1 = 23 ✓, 23 × 2 + 1 = 47 ✓. The rule is: multiply the previous term by 2, then add 1. For the 6th term: 47 × 2 + 1 = 95. This is a two-step pattern that combines multiplication and addition. When simple subtraction or division doesn't reveal a single constant, try combining operations.

SUMMARY

Lesson Summary

Summary

A pattern is a sequence of numbers that follows a rule. In an arithmetic sequence, you add (or subtract) the same common difference each time. In a geometric sequence, you multiply (or divide) by the same common ratio each time. To find the type, subtract or divide consecutive terms and check if the result stays constant.

Use the formula a(n) = a(1) + (n − 1) × d for arithmetic sequences and a(n) = a(1) × r⁽ⁿ⁻¹⁾ for geometric sequences to jump straight to any term. Some patterns use two-step rules or repeating cycles. Always check your answer by listing a few terms to make sure the rule works. These skills connect directly to linear and exponential functions in high school algebra.