Find Missing Terms in a Sequence - ISEE Middle Level Mathematics Achievement

SECTION 1

Why Do We Study Sequences?

People have been fascinated by number patterns for thousands of years. Ancient mathematicians in Egypt, India, and Greece noticed that certain numbers follow predictable rules. Understanding these rules helped them build pyramids, track the stars, and solve everyday problems.

A sequence is simply an ordered list of numbers that follow a rule. Think of it like a playlist where every song fits a theme. Once you know the theme, you can guess what comes next — or fill in a song that's missing.

~300 BC • Euclid's Elements

The Greek mathematician Euclid studied number patterns, including even numbers and prime numbers, in his famous book.

~600 AD • Indian Mathematicians

Scholars in India explored arithmetic sequences (adding the same number) and used them in astronomy and trade calculations.

1202 • Fibonacci's Liber Abaci

Leonardo Fibonacci published his famous rabbit-breeding sequence: 1, 1, 2, 3, 5, 8, 13… Each term is the sum of the two before it.

Today • ISEE & Real Life

Sequences appear on standardized tests like the ISEE. They also power computer algorithms, music, and even the spirals in sunflowers!

On the ISEE, you will see a list of numbers with one or more blanks. Your job is to figure out the rule and fill in the missing number. Let's learn exactly how to do that!

SECTION 2

Core Principles of Sequences

Before you can find a missing term, you need to understand a few key ideas. These ideas are your toolbox for cracking any sequence problem on the ISEE.

Term

Each number in a sequence is called a term. In the sequence 3, 7, 11, the first term is 3, the second term is 7, and so on.

Common Difference

When you add (or subtract) the same number each time, that number is the common difference. Example: 2, 5, 8, 11 → the common difference is +3.

Common Ratio

When you multiply (or divide) by the same number each time, that number is the common ratio. Example: 3, 6, 12, 24 → the common ratio is ×2.

Pattern Rule

The pattern rule is the instruction that tells you how to get from one term to the next. Finding this rule is the key to solving the problem.

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KEY TAKEAWAY

Think of a sequence like climbing stairs. If every step is the same height, you're adding the same number each time (common difference). If every step doubles in height, you're multiplying each time (common ratio). Once you figure out the step size, you can predict any stair — even one you can't see!

SECTION 3

Seeing the Pattern

The best way to understand sequences is to see them. The diagram below shows an arithmetic sequence (a sequence with a common difference) and a geometric sequence (a sequence with a common ratio) side by side.

The top row shows an arithmetic sequence where we add 4 each time. The dashed box is the missing term (11). The bottom row shows a geometric sequence where we multiply by 2 each time. The missing term is 40.

Notice how the arrows between the boxes show the rule. For the arithmetic sequence on top, every arrow says "+4." For the geometric sequence on the bottom, every arrow says "×2." On the ISEE, your first step should always be to figure out what goes on those arrows.

SECTION 4

The Math Behind Finding Missing Terms

Let's look at the simple formulas you can use. Don't worry — these aren't complicated! They just put into symbols what you already know from the diagrams.

ARITHMETIC SEQUENCE RULE

next term = current term + d

Here, d stands for the common difference. You find d by subtracting any term from the term that comes right after it. Example: if the sequence is 5, 9, 13, then d = 9 − 5 = 4.

GEOMETRIC SEQUENCE RULE

next term = current term × r

Here, r stands for the common ratio. You find r by dividing any term by the term that comes right before it. Example: if the sequence is 2, 6, 18, then r = 6 ÷ 2 = 3.

FINDING d (COMMON DIFFERENCE)

d = any term − the term before it

Always check with at least two pairs of terms to make sure the difference is consistent. If it's not the same each time, the sequence might be geometric or use a different rule.

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ISEE Test Tip

If the differences between terms are NOT the same, try dividing instead. If each term divided by the previous one gives the same number, you have a geometric sequence. If neither works, look for other patterns like adding increasing amounts (+1, +2, +3…) or perfect squares (1, 4, 9, 16…).

SECTION 5

Types of Sequences You'll See on the ISEE

The ISEE doesn't just test one kind of sequence. Here are the main types you should be ready for, along with how to spot them quickly.

This chart shows six types of sequences you may see on the ISEE. The most common are arithmetic and geometric. The strategy box at the bottom reminds you to start by finding differences, then try ratios if needed.

On the ISEE, most sequence questions will be arithmetic or geometric. However, perfect squares and growing differences also appear. The 3-step strategy at the bottom of the diagram works for almost every sequence problem you'll face.

SECTION 6

Worked Example: Step by Step

Let's walk through a complete problem just like one you'd see on the ISEE. Follow each step carefully.

Problem: What is the missing term in the sequence 5, 11, __, 23, 29?

Finding the Missing Term

1
Step 1 — Find the Differences You Can See

Look at pairs of terms that are next to each other. We can see that 11 − 5 = 6 and 29 − 23 = 6. Both differences are 6.

Common difference d = 6
2
Step 2 — Confirm the Pattern

Since the differences we found are both 6, this looks like an arithmetic sequence. If the missing term is correct, 23 − (missing term) should also equal 6.

Pattern confirmed: +6 each time
3
Step 3 — Apply the Rule

The missing term comes right after 11. So we calculate: 11 + 6 = 17.

Missing term = 17
4
Step 4 — Double-Check

Read the full sequence: 5, 11, 17, 23, 29. Check every gap: 11 − 5 = 6 ✓, 17 − 11 = 6 ✓, 23 − 17 = 6 ✓, 29 − 23 = 6 ✓. Every difference is 6. Our answer is correct!

Answer: 17 ✓

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ISEE Test Tip

Always double-check by plugging your answer back into the sequence. On the ISEE, wrong answer choices are designed to trick you, and a quick check can catch careless mistakes. This takes only a few seconds and is totally worth it!

SECTION 7

Strategies, Strengths, and Common Mistakes

Knowing the math is important, but knowing the right strategy can save you valuable time on test day. Let's compare different approaches and highlight common mistakes.

Strategy	When to Use It	Watch Out For
Subtract pairs	Try this first on every sequence problem. Works for arithmetic sequences.	If the differences aren't equal, don't force it — switch to dividing.
Divide pairs	When subtraction gives unequal results. Works for geometric sequences.	Make sure you divide each term by the one before it, not the other way around.
Second differences	When first differences grow by a steady amount (like +1, +2, +3…).	Takes an extra step. Stay organized — write your work down.
Plug in answers	When you're stuck and the answer choices are simple numbers.	This works but is slow. Use it as a backup, not your first move.

Comparison of strategies for finding missing terms

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KEY TAKEAWAY

Think of solving a sequence problem like being a detective. Your first clue is the differences between terms. If that clue doesn't crack the case, try ratios. If that doesn't work either, look at how the differences themselves change. With three tools in your belt, you can handle any sequence the ISEE throws at you.

SECTION 8

Looking Ahead: Sequences Beyond the ISEE

The sequence skills you're learning now are the foundation for bigger ideas in high school and college math. Here's a quick look at how what you're doing now connects to what's coming later.

What You Learn Now	Where It Leads Later
Finding common differences	Arithmetic series (adding up all the terms) and linear equations in algebra
Finding common ratios	Geometric series, exponential growth, and compound interest in finance
Spotting growing differences	Quadratic functions and parabolas in Algebra 2
Writing pattern rules	Writing explicit formulas and recursive formulas in precalculus

How ISEE sequence skills connect to future math

You don't need to know any of this advanced material for the ISEE. But it's cool to see that the skills you're building right now will help you for years to come. Mastering sequences now gives you a real head start!

SECTION 9

Practice Problems

Now it's your turn! Try these five problems. They go from easier to harder, just like on the real test. Remember: find the rule first, then apply it.

Question 1: CONCEPTUAL

What is the missing term in the sequence: 10, 15, 20, __, 30?

A. 22
B. 24
C. 25
D. 28

Show answer (C)

The correct answer is C (25). The common difference is 15 − 10 = 5, and 20 − 15 = 5. So the missing term is 20 + 5 = 25. Check: 30 − 25 = 5 ✓. Choice A (22) would mean a difference of 2, which doesn't match. Choice B (24) would mean a difference of 4, which doesn't match. Choice D (28) would mean a difference of 8, then only 2 to reach 30, which is inconsistent.

Question 2: BASIC CALCULATION

What is the missing term in the sequence: 3, 6, 12, __, 48?

A. 18
B. 24
C. 30
D. 36

Show answer (B)

The correct answer is B (24). This is a geometric sequence. Divide: 6 ÷ 3 = 2, and 12 ÷ 6 = 2, so the common ratio is 2. The missing term is 12 × 2 = 24. Check: 24 × 2 = 48 ✓. Choice A (18) is what you'd get by adding 6 each time, but the differences aren't constant (3, 6, 12 differ by 3, then 6). Choice C (30) comes from adding 18, which has no consistent pattern. Choice D (36) comes from multiplying by 3 instead of 2, but 12 ÷ 6 = 2 not 3.

Question 3: INTERMEDIATE

What is the missing term in the sequence: 50, 43, __, 29, 22?

A. 34
B. 36
C. 37
D. 39

Show answer (B)

The correct answer is B (36). This is a decreasing arithmetic sequence. Find the common difference: 43 − 50 = −7, and 22 − 29 = −7. So we subtract 7 each time. The missing term is 43 − 7 = 36. Check: 36 − 7 = 29 ✓. Choice A (34) would mean subtracting 9 then 5, which is inconsistent. Choice C (37) would mean subtracting 6, but the other differences are 7. Choice D (39) would mean subtracting 4 then 10, which doesn't match.

Question 4: APPLIED

A baker makes cookies for a bake sale. On Monday she made 2 cookies, on Tuesday she made 6 cookies, on Wednesday she made 12 cookies, and on Thursday she made 20 cookies. If the pattern continues, how many cookies will she make on Friday?

A. 24
B. 28
C. 30
D. 32

Show answer (C)

The correct answer is C (30). First find the differences: 6 − 2 = 4, 12 − 6 = 6, 20 − 12 = 8. The differences aren't equal, but they form their own pattern: 4, 6, 8 — they increase by 2 each time. So the next difference should be 10. Friday's total: 20 + 10 = 30. Choice A (24) adds only 4. Choice B (28) adds only 8 (repeating Thursday's difference instead of increasing it). Choice D (32) adds 12, skipping ahead too far in the difference pattern.

Question 5: CRITICAL THINKING

What is the missing term in the sequence: 1, 3, 7, __, 31, 63?

A. 13
B. 15
C. 17
D. 21

Show answer (B)

The correct answer is B (15). Look at the differences: 3 − 1 = 2, 7 − 3 = 4, and at the end 63 − 31 = 32. The differences are 2, 4, ?, ?, 32 — each one doubles! So the differences are 2, 4, 8, 16, 32. The missing term is 7 + 8 = 15. Check: 15 + 16 = 31 ✓ and 31 + 32 = 63 ✓. Another way to see it: each term is one less than a power of 2 (2−1=1, 4−1=3, 8−1=7, 16−1=15, 32−1=31, 64−1=63). Choice A (13) comes from adding 6 to 7, which doesn't fit the doubling pattern. Choice C (17) comes from adding 10, and Choice D (21) comes from adding 14 — neither fits.

SUMMARY

Putting It All Together

Summary

A sequence is an ordered list of numbers that follows a rule. To find a missing term, start by computing the differences between consecutive terms. If the differences are the same, you have an arithmetic sequence with a common difference. If the differences aren't equal, try dividing consecutive terms. If the quotient (result of division) is the same each time, you have a geometric sequence with a common ratio.

For trickier sequences, look at second differences (the differences of the differences) or check for perfect squares and alternating patterns. Always double-check your answer by plugging it back into the sequence and verifying the rule holds. On the ISEE, answer every question — even if you're unsure, use process of elimination to narrow down the choices and make your best guess. You've got this!