Find a Missing Term in a Sequence - ISEE Middle Level Quantitative Reasoning

SECTION 1

Why Do We Study Sequences?

Humans have been fascinated by number patterns for thousands of years. A sequence is simply a list of numbers that follow a rule. Ancient mathematicians noticed that patterns in numbers could help them predict eclipses, build pyramids, and even compose music.

Understanding sequences isn't just about math class. Today, sequences show up in computer coding, sports statistics, and even the way your favorite songs are structured. On the ISEE, you'll need to find patterns quickly and accurately.

~300 BCE • Euclid's Elements

The Greek mathematician Euclid studied number patterns and wrote about them in his famous book. He explored sequences of prime numbers and perfect squares.

~600 CE • Indian Mathematicians

Scholars in India discovered patterns in numbers that helped them develop our modern number system, including the concept of zero.

1202 • Fibonacci's Rabbit Problem

Leonardo Fibonacci introduced a famous sequence (1, 1, 2, 3, 5, 8, 13…) based on a puzzle about breeding rabbits. This pattern appears throughout nature.

1700s • Arithmetic & Geometric Series

Mathematicians like Carl Friedrich Gauss developed formulas that let us quickly find any term in a sequence without listing every number.

Today • Sequences on the ISEE

The ISEE tests your ability to recognize patterns. These questions appear in both the standard word-problem format and quantitative comparison format.

The big question every sequence problem asks is: What is the rule, and what number comes next (or fills the gap)? Let's learn how to crack that code.

SECTION 2

Core Principles of Sequences

Before you can find a missing term, you need to understand the main types of sequences you'll see on the ISEE. Each type follows a different kind of rule. Once you know the type, you can unlock the pattern.

Arithmetic Sequences

Each term is found by adding (or subtracting) the same number every time. Example: 3, 7, 11, 15 (add 4 each time).

Geometric Sequences

Each term is found by multiplying (or dividing) by the same number every time. Example: 2, 6, 18, 54 (multiply by 3 each time).

Perfect Square Sequences

The terms are the squares of counting numbers: 1, 4, 9, 16, 25. The differences between terms grow, but the pattern is that each term equals n × n.

Repeating (Cyclic) Patterns

A group of numbers repeats over and over. Example: 2, 5, 8, 2, 5, 8, 2, 5, 8… The block "2, 5, 8" keeps cycling.

Two-Operation Sequences

The rule involves two steps, like "multiply by 2 then add 1." Example: 1, 3, 7, 15 (double and add 1 each time).

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KEY TAKEAWAY

Think of a sequence like a playlist that follows a rule. If every song is exactly 30 seconds longer than the last, you have an arithmetic sequence. If every song is twice as long as the last, you have a geometric sequence. Your first job is always to figure out which "playlist rule" is being used!

SECTION 3

See the Pattern

A picture can make patterns much easier to see. The diagram below shows an arithmetic sequence where we add 5 each time. Notice how the arrows between terms always show the same jump. When a term is missing, you can use the arrows to figure out what belongs in the gap.

Each box holds a term in the sequence. The arrows show the common difference of +5. The dashed yellow box is the missing term. Since 8 + 5 = 13, the answer is 13.

Here's a quick strategy for the ISEE: always look at the terms around the gap. In the diagram above, 8 comes before the gap and 18 comes after. The difference between 8 and 18 is 10, and there are two jumps to get from 8 to 18. So each jump is 10 ÷ 2 = 5. That confirms the missing term is 8 + 5 = 13.

SECTION 4

The Math Behind Sequences

You don't need fancy formulas for the ISEE, but knowing a few simple rules will make you faster and more confident. Let's look at the key formulas for the two most common sequence types.

ARITHMETIC SEQUENCE RULE

Next term = Previous term + d

Here, d stands for the common difference (the amount you add or subtract each time). Find d by subtracting any term from the one after it.

GEOMETRIC SEQUENCE RULE

Next term = Previous term × r

Here, r stands for the common ratio (the number you multiply by each time). Find r by dividing any term by the one before it.

FINDING d WHEN A TERM IS MISSING

d = (Later term − Earlier term) ÷ (Number of jumps between them)

If you know the 2nd term is 8 and the 4th term is 18, there are 2 jumps. So d = (18 − 8) ÷ 2 = 5.

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ISEE Test Tip

On the ISEE, always check your answer by plugging it back into the sequence. If the pattern works with your answer in the gap, you're correct! This takes just a few seconds and catches careless mistakes.

SECTION 5

Types of Sequence Problems on the ISEE

The ISEE can ask about sequences in several ways. Let's look at the most common problem setups so you'll recognize them immediately on test day. The diagram below compares the two most tested types: arithmetic and geometric.

This side-by-side comparison shows arithmetic sequences (add/subtract) on the left and geometric sequences (multiply/divide) on the right. Use subtraction to check for arithmetic patterns first. If the differences change, try division to check for geometric patterns.

Problem Setup	What They Ask	Your First Move
A, B, ?, D, E	Find the missing middle term	Subtract consecutive known terms to find d
A, B, C, D, ?	Find the next term	Find the pattern, then apply it one more time
?, B, C, D, E	Find the first term	Find d, then work backward (subtract d from B)
Column A vs. Column B	Compare two sequence-related values	Find the missing terms in both columns, then compare

Common ways the ISEE asks about missing terms

SECTION 6

Worked Example: Step by Step

Let's walk through a full problem just like you'd see on the ISEE. Follow each step carefully, and notice how we check our work at the end.

Find the missing term: 5, 12, ?, 26, 33

1
Step 1 — Look at the known terms

We have the sequence 5, 12, ?, 26, 33. The missing term is the third one. Let's use the terms we know to find the pattern.
2
Step 2 — Find the differences between known consecutive terms

Start with terms that are next to each other. We know the 4th term (26) and the 5th term (33). The difference is 33 − 26 = 7. We also know the 1st term (5) and the 2nd term (12). The difference is 12 − 5 = 7.

Common difference d = 7
3
Step 3 — Confirm it's arithmetic

Both differences we found are the same (7), so this is an arithmetic sequence with d = 7. The rule is: each term equals the previous term plus 7.
4
Step 4 — Fill in the missing term

The 2nd term is 12. Add the common difference: 12 + 7 = 19. The missing 3rd term is 19.

Missing term = 19
5
Step 5 — Check your work

Read the full sequence: 5, 12, 19, 26, 33. Check each difference: 12 − 5 = 7 ✓, 19 − 12 = 7 ✓, 26 − 19 = 7 ✓, 33 − 26 = 7 ✓. Every gap is 7, so 19 is correct!

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Process of Elimination Tip

On the ISEE, answer choices are usually in order from least to greatest. Once you find d = 7, you know the missing term must be 12 + 7 = 19. You can immediately scan the answer choices for 19 without even checking the other choices. But if you have time, always verify!

SECTION 7

Strategies and Common Mistakes

Knowing the math is important, but smart test-taking strategies can save you time and help you avoid traps. Here are the top strategies and mistakes to watch out for on ISEE sequence problems.

Strategy ✓	Common Mistake ✗
Always find the difference between consecutive terms first	Guessing the pattern from just the first two terms
If differences change, try dividing consecutive terms to check for geometric	Assuming every sequence is arithmetic
Count the number of "jumps" between known terms carefully	Confusing the number of terms with the number of jumps (there's always one fewer jump than terms)
Plug your answer back in and verify the whole sequence	Picking an answer without checking that the full pattern works
For quantitative comparisons, compute both columns before comparing	Trying to compare columns without actually finding the missing terms

Do this, not that!

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REMEMBER

Think of finding a missing term like filling in a blank in a sentence. If the sentence is "I wake up, eat breakfast, _____, go to school," the blank is obviously "get dressed." In the same way, a sequence has a logical order. Once you know the rule, the missing term becomes obvious. Always answer every question on the ISEE — there's no penalty for wrong answers, so even a guess is better than leaving it blank!

SECTION 8

Connection to Bigger Ideas

The skills you're building with sequences connect to bigger ideas you'll see in high school math and beyond. Understanding how these fit together helps you see why this topic matters.

What You Learn Now	Where It Leads
Arithmetic sequences (add the same number)	Linear functions and graphing straight lines (y = mx + b)
Geometric sequences (multiply by the same number)	Exponential growth and decay (compound interest, population growth)
Finding patterns in number lists	Algebra: writing formulas to describe patterns
Working backward to fill in gaps	Solving equations by using inverse operations

From ISEE prep to future math success

On the ISEE, you might also see sequences that involve perfect squares (1, 4, 9, 16, 25…) or perfect cubes (1, 8, 27, 64…). These aren't arithmetic or geometric, but you can recognize them if you know your multiplication tables well. Keep practicing your times tables — they'll make you faster on sequence problems!

SECTION 9

Practice Problems

Time to put your skills to the test! These five problems go from easy to challenging. For each one, find the pattern, then find the answer. Remember: on the ISEE, always answer every question — no points are taken off for wrong answers.

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

The correct answer is (C) 25. This is an arithmetic sequence with a common difference of 5 (15 − 10 = 5, 20 − 15 = 5). The missing term is 20 + 5 = 25. We can verify: 30 − 25 = 5 ✓. Choice (A) 22 is wrong because 22 − 20 = 2, not 5. Choice (B) 24 is wrong because 24 − 20 = 4, not 5. Choice (D) 28 is wrong because 28 − 20 = 8, not 5.

Question 2: BASIC CALCULATION

What is the missing term in the sequence 3, 6, 12, ?, 48? (A) 18 (B) 24 (C) 30 (D) 36

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The correct answer is (B) 24. The differences between terms are 3, 6, 12 — they aren't constant, so this is not arithmetic. Checking ratios: 6 ÷ 3 = 2, 12 ÷ 6 = 2. This is a geometric sequence with a common ratio of 2. The missing term is 12 × 2 = 24. Check: 48 ÷ 24 = 2 ✓. Choice (A) 18 would mean 18 ÷ 12 = 1.5, not 2. Choice (C) 30 would mean 30 ÷ 12 = 2.5, not 2. Choice (D) 36 would mean 36 ÷ 12 = 3, not 2.

Question 3: INTERMEDIATE

In the sequence 7, ?, 23, 31, 39, what is the missing second term? (A) 12 (B) 14 (C) 15 (D) 16

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The correct answer is (C) 15. First, find the common difference using two consecutive known terms: 31 − 23 = 8, and 39 − 31 = 8. So d = 8. Working backward from the 3rd term: 23 − 8 = 15. Check: 15 − 7 = 8 ✓ and 23 − 15 = 8 ✓. The full sequence is 7, 15, 23, 31, 39. Choice (A) 12 is wrong because 12 − 7 = 5, not 8. Choice (B) 14 is wrong because 14 − 7 = 7, not 8. Choice (D) 16 is wrong because 16 − 7 = 9, not 8.

Question 4: APPLIED

Quantitative Comparison: The sequence 2, 6, 18, 54, 162, ... is a geometric sequence. Column A: The 6th term of the sequence Column B: 500 (A) Column A is greater. (B) Column B is greater. (C) The two quantities are equal. (D) Cannot be determined.

Show answer

The correct answer is (B) Column B is greater. The common ratio is 6 ÷ 2 = 3. Multiply each term by 3: 2, 6, 18, 54, 162, 486. The 6th term is 486. Column A = 486 and Column B = 500. Since 486 < 500, Column B is greater.

Question 5: CRITICAL THINKING

Quantitative Comparison: Sequence P: 4, 11, 18, 25, 32, ... Sequence Q: 100, 93, 86, 79, 72, ... Column A: The 8th term of Sequence P Column B: The 8th term of Sequence Q (A) Column A is greater. (B) Column B is greater. (C) The two quantities are equal. (D) Cannot be determined.

Show answer

The correct answer is (A) Column A is greater. Sequence P is arithmetic with a common difference of +7 (11 − 4 = 7). Using the formula: nth term = first term + d × (n − 1), the 8th term of P = 4 + 7 × 7 = 4 + 49 = 53. Sequence Q is arithmetic with a common difference of −7 (93 − 100 = −7). The 8th term of Q = 100 + (−7) × 7 = 100 − 49 = 51. You can verify by listing all terms: Sequence P: 4, 11, 18, 25, 32, 39, 46, 53 Sequence Q: 100, 93, 86, 79, 72, 65, 58, 51 Since 53 > 51, Column A is greater. Choice (B) is wrong because 53 > 51, not less. Choice (C) is wrong because 53 ≠ 51. Choice (D) is wrong because both sequences are fully defined, giving exact values.

SUMMARY

Putting It All Together

Summary

A sequence is a list of numbers that follows a rule. To find a missing term, start by checking whether the sequence is arithmetic (same difference between terms) or geometric (same ratio between terms). Subtract consecutive known terms first. If the differences are the same, you've found the common difference (d). If not, try dividing to find the common ratio (r).

Once you know the rule, apply it to fill the gap. If the missing term is in the middle, use the terms on either side to find it. Always check your answer by reading through the complete sequence. For quantitative comparisons, find the missing terms in both columns before choosing your answer. And remember: never leave a question blank on the ISEE — there's no penalty for guessing!