Find a Missing Number in an Arithmetic Sequence - ISEE Lower Level Mathematics Achievement

SECTION 1

Where Do Number Patterns Come From?

People have loved number patterns for thousands of years! Long ago, mathematicians noticed that some numbers follow a special rule. When you add the same amount each time, you get what we call an arithmetic sequence (say it like: air-ith-MET-ick). That just means a list of numbers where you keep adding the same number.

~2000 BC • Ancient Babylonians

People in ancient Babylon used number patterns to track crops and trade goods. They carved patterns into clay tablets!

~500 BC • Pythagoras & the Greeks

Greek mathematicians studied number patterns and called them beautiful. They believed patterns were the key to understanding the world.

~1800s • Carl Friedrich Gauss

As a young student, Gauss amazed his teacher by quickly adding 1 + 2 + 3 all the way to 100. He used a pattern shortcut!

Today • Patterns Are Everywhere

Number patterns show up on tests like the ISEE. They also help scientists, builders, and even game designers!

On the ISEE, you will see a list of numbers with one missing. Your job is to figure out the pattern and fill in the blank. Let's learn how!

SECTION 2

Core Ideas You Need to Know

Before we solve problems, let's learn four big ideas. These are your superpowers for finding missing numbers!

Sequence

A sequence is just a list of numbers in order. Example: 3, 6, 9, 12.

Common Difference

The common difference is the number you add each time. In 3, 6, 9, 12, you add 3 every time.

The Rule

Every arithmetic sequence follows a rule. The rule tells you what to add (or subtract) to get the next number.

Find the Gap

To find a missing number, subtract two neighbors. The answer is your common difference. Then use it to fill in the blank!

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KEY TAKEAWAY

Think of an arithmetic sequence like climbing stairs. Each step is the same height. If one stair is hidden, you can figure out how tall it is because all the stairs are equal! The common difference is the height of each stair.

SECTION 3

See the Pattern!

Let's look at the sequence 4, 7, 10, ?, 16. The picture below shows how the numbers grow by jumping the same amount each time.

Each curved arrow shows a jump of +3. The yellow question mark is where the missing number goes. Since 10 + 3 = 13, the answer is 13.

See how every jump is the same size? That's the secret of an arithmetic sequence. Once you find the jump, you can fill in any missing number!

SECTION 4

The Three-Step Method

Here is a simple three-step method you can use every time. It works like a recipe!

STEP 1 — FIND THE COMMON DIFFERENCE

Common Difference = Next Number − Previous Number

Pick any two numbers that are right next to each other. Subtract the smaller one from the bigger one. The answer is your common difference.

STEP 2 — CHECK YOUR DIFFERENCE

Try it between another pair of neighbors

Make sure the same difference works between other pairs. This double-checks your work!

STEP 3 — USE THE DIFFERENCE

Missing Number = Number Before It + Common Difference

Add the common difference to the number just before the blank. Or subtract the common difference from the number just after the blank. Either way works!

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ISEE Test Tip

On the ISEE, there is no penalty for guessing! If you can't find the pattern right away, try each answer choice. Plug it in and see if the jumps stay the same. Always answer every question.

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Different Kinds of Patterns

On the ISEE, the missing number can be anywhere — at the beginning, in the middle, or at the end. The pattern might go up (adding) or go down (subtracting). Let's see all the types!

The missing number can appear in three positions. For Type 1, add. For Type 2, add from the number before. For Type 3, subtract from the number after.

Notice something cool? For Type 3, you subtract to go backward. It's like rewinding! But the common difference is still the same number.

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Let's Solve One Together

Here is a problem just like you might see on the ISEE. Let's solve it step by step!

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Problem

What number is missing in this sequence? 6, 11, 16, __, 26, 31

Step-by-Step Solution

1
Step 1 — Find the Common Difference

Look at the first two numbers: 11 − 6 = 5. The common difference might be 5.

Possible difference = 5
2
Step 2 — Check With Another Pair

Try the next pair: 16 − 11 = 5. Great, it matches! Also check: 31 − 26 = 5. Perfect!

Confirmed: common difference = 5
3
Step 3 — Fill in the Blank

The number before the blank is 16. Add the common difference: 16 + 5 = 21.

Missing number = 21
4
Step 4 — Double-Check

Read the full sequence: 6, 11, 16, 21, 26, 31. Each jump is +5. It works!

✓ Answer: 21

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REMEMBER THIS!

Always check your answer by reading the whole sequence out loud. Does every jump feel the same? If yes, you nailed it!

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Tips and Common Traps

Even great math students can make mistakes on pattern problems. Here are some tips to help you and traps to avoid.

✅ Do This	❌ Avoid This
Check the difference between at least two pairs of neighbors.	Only checking one pair — you might pick the wrong difference.
Read the full sequence with your answer plugged in.	Picking an answer without double-checking.
Use subtraction to go backward when the blank is at the start.	Adding when you should subtract.
Use process of elimination — try each answer choice if stuck.	Leaving the question blank — there's no penalty for guessing!

Smart strategies for missing-number problems

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TEST STRATEGY

If the numbers in the sequence are getting bigger, you are adding. If they are getting smaller, you are subtracting. Look at the direction first — it helps you know what to do right away!

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Patterns Beyond Adding

On the ISEE, most sequences use adding or subtracting. But as you grow as a math student, you will see fancier patterns too. Here's a sneak peek!

Type	What Happens	Example
Adding (arithmetic)	Add the same number each time	2, 5, 8, 11, 14 (+3)
Subtracting	Subtract the same number each time	30, 25, 20, 15, 10 (−5)
Multiplying (future topic)	Multiply by the same number each time	2, 4, 8, 16, 32 (×2)

Types of number patterns

For now, focus on adding and subtracting patterns. That's what the ISEE Lower Level tests. You've got this!

SECTION 9

Practice Problems

Time to practice! Try each problem on your own, then check the answer. Remember: find the common difference first, then use it to find the missing number.

Question 1: CONCEPTUAL

What is the missing number in this sequence? 2, 4, 6, 8, __

A. 9
B. 10
C. 11
D. 12

Show answer (B)

The correct answer is B (10). The common difference is 2 because 4 − 2 = 2 and 6 − 4 = 2. So 8 + 2 = 10. Choice A (9) adds only 1. Choice C (11) adds 3. Choice D (12) adds 4. None of those match the pattern of adding 2.

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 correct answer is C (20). The common difference is 5 because 10 − 5 = 5 and 15 − 10 = 5. So 15 + 5 = 20. You can also check: 25 − 20 = 5. It works! Choice A (16) adds only 1. Choice B (18) adds 3. Choice D (22) adds 7. Only 20 keeps the same +5 jump.

Question 3: INTERMEDIATE

What is the missing number in this sequence? __, 18, 24, 30, 36

A. 10
B. 12
C. 14
D. 16

Show answer (B)

The correct answer is B (12). The common difference is 6 because 24 − 18 = 6 and 30 − 24 = 6. Since the blank is at the beginning, subtract: 18 − 6 = 12. Choice A (10) would give a jump of 8 from 10 to 18. Choice C (14) would give a jump of 4. Choice D (16) would give a jump of 2. Only 12 makes every jump equal to 6.

Question 4: APPLIED

Maria saves money each week. She saved $3 in Week 1, $7 in Week 2, $11 in Week 3, and $15 in Week 4. If she keeps saving the same extra amount each week, how much will she save in Week 5?

A. $17
B. $18
C. $19
D. $20

Show answer (C)

The correct answer is C ($19). The sequence is 3, 7, 11, 15, and the common difference is 4 because 7 − 3 = 4 and 11 − 7 = 4. So Week 5 is 15 + 4 = 19. Choice A ($17) adds only 2. Choice B ($18) adds 3. Choice D ($20) adds 5. The pattern adds 4 each week, so $19 is correct.

Question 5: CRITICAL THINKING

What is the missing number in this sequence? 8, 14, 20, __, 32

A. 22
B. 24
C. 26
D. 28

Show answer (C)

The correct answer is C (26). This sequence goes up, so you're adding! The common difference is 6 because 14 − 8 = 6 and 20 − 14 = 6. So 20 + 6 = 26. You can check: 26 + 6 = 32. It works! Choice A (22) would mean adding 2 then 10, which isn't the same. Choice B (24) would mean adding 4 then 8. Choice D (28) would mean adding 8 then 4. Only 26 keeps the pattern of adding 6 every time.

SUMMARY

Lesson Summary

Summary

An arithmetic sequence is a list of numbers where you add (or subtract) the same amount each time. That amount is called the common difference. To find a missing number, use the three-step method: first, find the common difference by subtracting two neighbors. Second, check it with another pair. Third, add or subtract to fill in the blank.

Remember that the blank can appear at the beginning, middle, or end of the sequence. If it's at the start, subtract the common difference from the next number. Always double-check by reading the whole sequence with your answer plugged in. On the ISEE, there is no penalty for guessing, so always answer every question. You've got this!