Predicting Later Terms — Predict a later term using the pattern rule. - SHSAT Math

SECTION 1

Why Do We Need to Predict Terms?

People have been fascinated by number patterns for thousands of years. Ancient mathematicians noticed that certain numbers follow a rule, and if you know the rule, you can figure out any number in the list — even the 100th or the 1,000th! A list of numbers that follows a rule is called a sequence. On the SHSAT, you'll often need to find a term far down the line without writing out every single term. That's exactly what this lesson teaches you.

~300 BCE • Euclid's Elements

The Greek mathematician Euclid studied patterns in numbers. He wrote about sequences of prime numbers and perfect numbers.

~600 CE • Indian Mathematics

Mathematicians in India developed formulas for arithmetic sequences. They used these formulas to solve real-world problems about trade and construction.

1700s • Carl Friedrich Gauss

As a young student, Gauss amazed his teacher by quickly adding the numbers 1 through 100. He used a pattern rule instead of adding one by one!

Today • Standardized Tests

Pattern-prediction questions appear on the SHSAT and many other exams. They test your ability to find a rule and use it efficiently.

The big question is: how do you jump straight to a later term without listing every number before it? That's the skill we'll build in this lesson.

SECTION 2

Core Ideas Behind Sequences

Before we start predicting later terms, let's nail down a few key ideas. Every sequence question on the SHSAT depends on these building blocks.

Term

Each number in a sequence is called a term. The first number is Term 1, the second is Term 2, and so on.

Position Number (n)

The position number tells you where a term sits in the sequence. We use the letter n for the position.

Common Difference (d)

In an arithmetic sequence, you add (or subtract) the same number each time. That number is the common difference, d.

Pattern Rule (Formula)

The pattern rule is a formula that connects the position number to the term value. It lets you skip ahead to any term you want.

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KEY TAKEAWAY

Think of a sequence like a vending machine. You press a button number (the position, n), and a snack pops out (the term value). The pattern rule is the machine's program — it turns any button number into the right snack. If you know the program, you can figure out what comes out of button 50 without pressing buttons 1 through 49 first!

SECTION 3

Seeing the Pattern

Let's look at a simple sequence and see how the position number connects to each term. In the diagram below, each circle is a term. The number on top is the position (n), and the number inside is the term's value.

This diagram shows the sequence 5, 8, 11, 14, 17, … with a common difference of 3. The pattern rule at the bottom lets you jump directly to the 20th term.

Notice how each term grows by exactly 3. That +3 is the common difference. The formula at the bottom is the magic shortcut. You plug in the position number (n), and out comes the term value. No need to count up one by one!

SECTION 4

The Formula for Predicting Terms

The most common type of sequence on the SHSAT is the arithmetic sequence (a sequence where you add or subtract the same number each time). Here is the formula you need to know.

ARITHMETIC SEQUENCE FORMULA

aₙ = a₁ + (n − 1) × d

aₙ = the term you want to find • a₁ = the first term • n = position number • d = common difference (the amount added each time)

Why does the formula use (n − 1) instead of just n? Think about it this way: the first term hasn't been changed at all — you haven't added d yet. The second term has d added once. The third term has d added twice. So the number of times you add d is always one less than the position number.

SIMPLIFIED FORM

aₙ = d × n + (a₁ − d)

If you distribute the multiplication, you get this version. It looks like a line equation (y = mx + b), which is useful for graphing.

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SHSAT Tip

Some sequences on the SHSAT use multiplication instead of addition. For example: 2, 6, 18, 54, … (each term is multiplied by 3). These are called geometric sequences. The formula is different: aₙ = a₁ × r⁽ⁿ⁻¹⁾, where r is the common ratio. Most SHSAT questions stick with arithmetic sequences, but stay alert!

SECTION 5

Recognizing Different Sequence Types

Before you can predict a later term, you need to figure out what kind of sequence you're looking at. Here's a comparison chart and a visual to help you tell them apart.

Feature	Arithmetic Sequence	Geometric Sequence
Rule	Add (or subtract) the same number	Multiply (or divide) by the same number
Example	4, 7, 10, 13, 16, …	3, 6, 12, 24, 48, …
Key Number	Common difference d = 3	Common ratio r = 2
Formula	aₙ = a₁ + (n − 1) × d	aₙ = a₁ × r⁽ⁿ⁻¹⁾
Growth	Steady (straight line on a graph)	Fast (curve on a graph)

Arithmetic vs. Geometric Sequences

The arithmetic sequence (cyan) forms a straight line because it grows by the same amount each step. The geometric sequence (pink) curves upward quickly because it multiplies each step.

On the SHSAT, always check the differences between terms first. If the differences are the same, it's arithmetic. If the ratios are the same, it's geometric. Once you know the type, you know which formula to use.

SECTION 6

Worked Example — Finding the 50th Term

Let's work through a full problem, step by step, just like you would on the SHSAT.

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PROBLEM

The first five terms of a sequence are: 7, 13, 19, 25, 31, … What is the 50th term?

Finding the 50th Term

1
Step 1 — Find the Common Difference

Subtract consecutive terms: 13 − 7 = 6, 19 − 13 = 6, 25 − 19 = 6. The difference is always 6, so this is an arithmetic sequence with d = 6.

d = 6
2
Step 2 — Identify the First Term

The first number in the sequence is 7, so a₁ = 7.

a₁ = 7
3
Step 3 — Write the Formula

Use the arithmetic sequence formula: aₙ = a₁ + (n − 1) × d. Plug in the values: aₙ = 7 + (n − 1) × 6.

aₙ = 7 + (n − 1) × 6
4
Step 4 — Substitute n = 50

Replace n with 50: a₅₀ = 7 + (50 − 1) × 6. Simplify inside the parentheses first: 50 − 1 = 49.

a₅₀ = 7 + 49 × 6
5
Step 5 — Calculate

Multiply: 49 × 6 = 294. Then add: 7 + 294 = 301.

a₅₀ = 301

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CHECK YOUR WORK

You can always check by plugging in a small position number you already know. Try n = 1: 7 + (1 − 1) × 6 = 7 + 0 = 7. That matches the first term! Try n = 3: 7 + (3 − 1) × 6 = 7 + 12 = 19. That matches too. Your formula is correct.

SECTION 7

Common Mistakes and How to Avoid Them

Even if you understand the formula, small mistakes can cost you points on test day. Here are the most common errors students make — and how to dodge them.

Common Mistake	Why It Happens	How to Fix It
Using n instead of (n − 1)	Students forget that you add d starting from the second term, not the first.	Always check: plug n = 1 into your formula. You should get a₁.
Wrong sign on d	If the sequence decreases (like 20, 17, 14, …), d is negative (−3), not positive.	Subtract: second term minus first term. Keep the sign!
Confusing arithmetic with geometric	Students use the addition formula when the pattern uses multiplication.	Check differences AND ratios between terms before choosing a formula.
Order of operations error	Students add a₁ + (n − 1) first, then multiply by d.	Multiply (n − 1) × d first, then add a₁. Parentheses and multiplication come before addition.

Connecting to Harder Sequence Questions

The arithmetic sequence formula is the foundation, but harder test questions might twist the idea. Here's a look at what you might see on the most challenging SHSAT questions — and how the same thinking still applies.

What You Know Now	What Comes Next
Find the nth term of an arithmetic sequence	Find the sum of the first n terms (Gauss's trick!)
Recognize a constant common difference	Handle sequences where the difference itself changes (quadratic sequences)
Use the formula aₙ = a₁ + (n − 1) × d	Write functions like f(x) = mx + b (linear functions are arithmetic sequences!)
Identify arithmetic vs. geometric	Work with other patterns: Fibonacci-type, alternating, or rule-based sequences

From Sequences to More Advanced Topics

Here's a cool connection: the arithmetic sequence formula aₙ = a₁ + (n − 1) × d is really just a linear function in disguise. If you expand it, you get aₙ = d × n + (a₁ − d). That's the same form as y = mx + b. The common difference d is the slope! So every time you find a later term, you're actually doing algebra with linear functions.

SECTION 9

Practice Problems

Try these five problems on your own. They start easy and get harder. After each one, check the answer and read the explanation.

Question 1: CONCEPTUAL

A sequence starts 10, 15, 20, 25, 30, … Is this arithmetic or geometric? What is the common difference?

Show answer

This is an arithmetic sequence because you add the same number each time. Subtract consecutive terms: 15 − 10 = 5, 20 − 15 = 5, 25 − 20 = 5. The common difference is d = 5. Each term is exactly 5 more than the previous one.

Question 2: BASIC CALCULATION

In the sequence 3, 8, 13, 18, 23, … what is the 30th term?

Show answer

First, find d: 8 − 3 = 5. The first term a₁ = 3. Use the formula: a₃₀ = 3 + (30 − 1) × 5 = 3 + 29 × 5 = 3 + 145 = 148. The 30th term is 148. You can check by plugging in n = 1: 3 + 0 × 5 = 3 ✓.

Question 3: INTERMEDIATE

The 1st term of an arithmetic sequence is 100 and the 5th term is 80. What is the 20th term?

Show answer

You know a₁ = 100 and a₅ = 80. Use the formula to find d: 80 = 100 + (5 − 1) × d → 80 = 100 + 4d → 4d = −20 → d = −5. Now find the 20th term: a₂₀ = 100 + (20 − 1) × (−5) = 100 + 19 × (−5) = 100 − 95 = 5. The 20th term is 5. Notice d is negative because the sequence decreases.

Question 4: APPLIED

Maria saves money every week. In week 1 she saves $12. Each week after that she saves $4 more than the previous week ($12, $16, $20, $24, …). How much does she save in week 15?

Show answer

This is an arithmetic sequence with a₁ = 12 and d = 4. The amount saved in week 15 is a₁₅ = 12 + (15 − 1) × 4 = 12 + 14 × 4 = 12 + 56 = 68. Maria saves $68 in week 15. The formula models real-world situations like savings, stacking objects, or any process that grows by a constant amount.

Question 5: CRITICAL THINKING

Two arithmetic sequences are given. Sequence A: 5, 11, 17, 23, … Sequence B: 200, 193, 186, 179, … At what position number n do the two sequences have the same value?

Show answer

For Sequence A: a₁ = 5, d = 6, so aₙ = 5 + (n − 1) × 6 = 6n − 1. For Sequence B: a₁ = 200, d = −7, so aₙ = 200 + (n − 1) × (−7) = 207 − 7n. Set them equal: 6n − 1 = 207 − 7n → 13n = 208 → n = 16. At position 16, both sequences equal 6(16) − 1 = 95. This problem combines sequence formulas with solving an equation — a common SHSAT challenge!

SUMMARY

Lesson Summary

Summary

A sequence is a list of numbers that follows a rule. In an arithmetic sequence, you add the same number (the common difference, d) each time. To predict any later term without listing them all, use the pattern rule formula: aₙ = a₁ + (n − 1) × d. Just plug in the position number (n) and the formula gives you the term.

Always follow these steps: find d by subtracting consecutive terms, identify a₁ (the first term), write the formula, plug in n, and simplify carefully using order of operations. Check your answer by testing a position you already know. Remember, the formula is really a linear function — a powerful connection you'll use again and again in algebra.