Find a missing term in a sequence. - ISEE Lower Level Quantitative Reasoning

SECTION 1

Why Do Patterns Matter?

People have loved patterns for thousands of years! A sequence is a list of numbers that follow a rule. Long ago, mathematicians noticed that patterns are everywhere — in nature, music, and buildings.

Think about counting by twos: 2, 4, 6, 8. That's a sequence! Once you know the rule, you can find any number in the list — even a missing one.

3000 BC • Ancient Patterns

People in ancient Egypt and Babylon used number patterns to build pyramids and track the stars.

500 BC • Greek Math Lovers

Greek mathematicians like Pythagoras studied number patterns and called them magical. They even made music using patterns!

1202 • Fibonacci's Famous Sequence

An Italian mathematician named Fibonacci shared a special sequence: 1, 1, 2, 3, 5, 8... Each number is the sum of the two before it!

Today • Patterns on the ISEE

The ISEE test asks you to find missing numbers in sequences. It's a skill you use every day!

So here's the big question: if someone hides one number in a pattern, can you figure out what it is? That's exactly what we'll learn today!

SECTION 2

Core Ideas About Sequences

Before we start finding missing numbers, let's learn a few important ideas. These will be your tools!

What Is a Sequence?

A sequence is a list of numbers in a special order. Each number is called a term.

What Is a Rule?

Every sequence follows a rule. The rule tells you how to get from one number to the next. For example, "add 3" or "multiply by 2."

Find the Pattern First

Always look at the numbers you DO have. Ask yourself: what happens between each pair of numbers? That's how you find the rule.

Check Your Work

After you find the missing number, plug it back in. Does the whole sequence follow the same rule? If yes, you've got it!

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KEY TAKEAWAY

Think of a sequence like a train. Each car is a number. The rule is like the connector between the cars. If one car is missing, you can look at the connectors on both sides to figure out which car belongs there!

SECTION 3

See the Pattern!

Let's look at a picture that shows how a sequence works. This diagram shows the sequence 3, 6, 9, ?, 15 and how to find the missing term.

Each box is a term in the sequence. The arrows show the rule — add 3 each time. The yellow box is the missing term: 12.

See how we subtract neighbors to find the rule? That trick works with most sequences on the ISEE. Always check that your answer fits both the number before AND the number after!

SECTION 4

The Three Main Types of Sequences

On the ISEE, you'll mostly see three types of sequences. Let's learn each one!

Type 1: Add (or Subtract) the Same Number

ADD THE SAME NUMBER

Next Term = Current Term + Same Number

Example: 5, 10, 15, 20 → the rule is add 5 each time.

Type 2: Multiply (or Divide) by the Same Number

MULTIPLY BY THE SAME NUMBER

Next Term = Current Term × Same Number

Example: 2, 4, 8, 16 → the rule is multiply by 2 each time.

Type 3: Two-Step Rules

TWO-STEP RULE

The amount added changes each time

Example: 1, 2, 4, 7, 11 → you add 1, then 2, then 3, then 4. The amount you add grows by 1 each time.

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ISEE Test Tip

If adding doesn't reveal a pattern, try subtracting or multiplying! On the ISEE, you won't be penalized for a wrong answer, so always guess if you're unsure. Use process of elimination — cross out answers that are clearly too big or too small.

SECTION 5

Your Step-by-Step Strategy

Here's a simple plan you can follow every time you see a sequence question on the ISEE. Think of it as your detective toolkit!

Follow these four steps every time. Step 4 is super important — always check your answer before moving on!

If subtraction doesn't show a pattern, try dividing each term by the one before it. If each time you get the same answer, the rule is multiply! For example, 3, 6, 12, 24 → each term is the one before it × 2.

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What If the Missing Number Is First?

Don't panic! Use the terms you DO have to find the rule. Then work backwards. If the sequence is ?, 10, 15, 20 and the rule is add 5, then the missing first term is 10 − 5 = 5.

SECTION 6

Worked Example

Let's work through a problem together, step by step. Ready? Here we go!

Find the missing term: 4, 10, ?, 22, 28

1
Step 1 — Find the Differences

Look at the numbers you know. From 4 to 10, the difference is 10 − 4 = 6. From 22 to 28, the difference is 28 − 22 = 6.

The difference is 6 each time.
2
Step 2 — Spot the Rule

Both differences are 6. So the rule is add 6 each time.

Rule: add 6
3
Step 3 — Fill the Gap

The term before the blank is 10. Apply the rule: 10 + 6 = 16.

Missing term = 16
4
Step 4 — Check Both Sides

Does 16 + 6 = 22? Yes! The full sequence is 4, 10, 16, 22, 28. Every step adds 6. ✓

Confirmed! The answer is 16.

Great job following along! See how checking both sides at the end gives you confidence? On the real ISEE, that quick check can save you from picking the wrong answer.

SECTION 7

Common Mistakes & How to Avoid Them

Even smart students make mistakes on sequence problems. Here are the most common ones — and how to dodge them!

Common Mistake	Why It Happens	How to Fix It
Only checking one pair of numbers	You look at only the first two terms and assume you know the rule.	Always check at least two pairs before deciding the rule.
Forgetting to check after the blank	You apply the rule to get the missing term but don't see if the NEXT term also follows the rule.	Always check both sides of the blank.
Mixing up add and multiply	The numbers get bigger fast, so you think it's adding, but it's actually multiplying.	If the differences aren't the same, try dividing each number by the one before it.
Rushing and making an arithmetic error	You know the rule but mess up the addition or subtraction.	Slow down! Write out your work in the test booklet.

Common mistakes on ISEE sequence problems

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REMEMBER

On the ISEE, there is NO penalty for guessing. If you're stuck, eliminate answers that look too big or too small, then pick from what's left. Never leave a question blank!

SECTION 8

Beyond Simple Sequences

Once you're comfortable with "add the same number" sequences, you might see trickier patterns. Here's how the basic idea connects to more advanced ones.

What You Know Now	What Comes Next
Add the same number (like +5)	Add a changing amount (like +1, +2, +3, +4)
Multiply by the same number (like ×2)	Patterns that switch between add and multiply
Sequences of whole numbers	Sequences with fractions or decimals
Numbers going up	Numbers going down (subtracting or dividing)

How sequence skills grow over time

For now, focus on the patterns we learned today. They cover most of what you'll see on the ISEE Lower Level. As you get more practice, the trickier patterns will feel easy too!

SECTION 9

Practice Problems

Time to try some on your own! Remember your detective plan: find the differences, spot the rule, fill the gap, and check. You've got this!

Question 1: CONCEPTUAL

What is the missing number in this sequence? 2, 4, 6, ?, 10

A. 7
B. 8
C. 9
D. 12

Show answer (B)

The rule is add 2 each time. 6 + 2 = 8, and 8 + 2 = 10, which matches! Choice A (7) would mean adding 1 from 6, which doesn't match the pattern. Choice C (9) would mean adding 3 from 6, which breaks the rule. Choice D (12) is way too big — it would mean adding 6 from 6.

Question 2: BASIC CALCULATION

What is the missing number in this sequence? 5, 10, 15, ?, 25

A. 16
B. 18
C. 20
D. 22

Show answer (C)

The rule is add 5 each time (10 − 5 = 5, 15 − 10 = 5). So the missing term is 15 + 5 = 20. Check: 20 + 5 = 25. ✓ Choice A (16) is only 1 more than 15. Choice B (18) means adding 3, not 5. Choice D (22) means adding 7 from 15, which doesn't fit.

Question 3: INTERMEDIATE

What is the missing number in this sequence? ?, 12, 18, 24, 30

A. 4
B. 6
C. 8
D. 10

Show answer (B)

First, find the rule using the terms you know: 18 − 12 = 6, 24 − 18 = 6, 30 − 24 = 6. The rule is add 6. Since 12 is the second term, work backwards: 12 − 6 = 6. The first term is 6. Choice A (4) would mean adding 8 to get to 12, not 6. Choice C (8) would mean adding 4 to get to 12. Choice D (10) would mean adding 2, which doesn't match.

Question 4: APPLIED

Maria is saving money each week. In week 1 she saved $3, in week 2 she saved $6, in week 3 she saved $12, and in week 4 she saved $24. How much will she save in week 5?

A. $30
B. $36
C. $48
D. $72

Show answer (C)

Let's check the differences: 6 − 3 = 3, 12 − 6 = 6, 24 − 12 = 12. The differences aren't the same, so try dividing: 6 ÷ 3 = 2, 12 ÷ 6 = 2, 24 ÷ 12 = 2. The rule is multiply by 2! So 24 × 2 = 48. Choice A ($30) adds 6 to 24. Choice B ($36) adds 12 to 24. Choice D ($72) multiplies by 3 instead of 2.

Question 5: CRITICAL THINKING

What is the missing number in this sequence? 1, 3, 6, ?, 15

A. 8
B. 9
C. 10
D. 12

Show answer (C)

This is a changing-difference pattern! From 1 to 3, the difference is 2. From 3 to 6, the difference is 3. The differences are going up by 1 each time (2, 3, 4, 5…). So the next difference is 4: 6 + 4 = 10. Check: the next difference should be 5, and 10 + 5 = 15. ✓ Choice A (8) would mean adding 2, which doesn't continue the growing pattern. Choice B (9) would mean adding 3 again, but the differences need to grow. Choice D (12) would mean adding 6, which skips ahead.

SUMMARY

Putting It All Together

Summary

A sequence is a list of numbers that follow a rule. To find a missing term, use the four-step detective plan: find the differences, spot the rule, fill the gap, and check both sides. Most ISEE sequences use adding, subtracting, or multiplying by the same number.

If the missing number is at the beginning, work backwards using the opposite operation. Watch out for changing-difference patterns where the amount added goes up each time. And remember — on the ISEE, there's no penalty for guessing, so always answer every question. Use process of elimination to cross out wrong answers. You've got this!