Missing Pattern Terms — Find a missing term in an arithmetic pattern. - SHSAT Math

SECTION 1

Where Do Patterns Come From?

Humans have been fascinated by number patterns for thousands of years. Long before calculators existed, people in ancient civilizations noticed that certain numbers follow a predictable rule. An arithmetic pattern (a list of numbers where you add or subtract the same amount each time) is one of the oldest ideas in math.

~2000 BCE • Babylonian Number Tables

Ancient Babylonians carved number sequences into clay tablets. They used patterns to track trade, measure land, and predict the seasons.

~300 BCE • Euclid's Elements

The Greek mathematician Euclid studied number relationships and proved properties of arithmetic sequences. His work laid a foundation for algebra.

1800s • Carl Friedrich Gauss

As a young student, Gauss amazed his teacher by quickly adding the numbers 1 through 100. He spotted an arithmetic pattern that let him solve it in seconds.

Today • Standardized Test Staple

The SHSAT and other admissions tests regularly ask you to find missing terms in arithmetic patterns. It is a skill that proves you can think logically.

So here is the big question: if someone gives you a list of numbers and one of them is hidden, how do you figure out the missing value? That is exactly what this lesson will teach you.

SECTION 2

Core Principles of Arithmetic Patterns

Before we solve any problems, let's nail down the key ideas. An arithmetic pattern is built on one simple rule: each term is created by adding (or subtracting) the same number every time. That number has a special name—the common difference.

Sequence

A sequence is an ordered list of numbers that follows a rule. Example: 4, 7, 10, 13.

Term

Each number in the list is called a term. The first number is the 1st term, the next is the 2nd term, and so on.

Common Difference

The common difference (d) is the constant amount added to get from one term to the next. It can be positive, negative, or even a fraction.

Missing Term

A missing term is a blank or unknown value somewhere in the sequence. You find it by using the common difference.

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KEY TAKEAWAY

Think of an arithmetic pattern like climbing stairs. Each step is the same height. If you know the height of one step (the common difference), you can figure out the height of any step—even one you can't see.

SECTION 3

See the Pattern on a Number Line

A number line is one of the best ways to see an arithmetic pattern. Each term sits on the line, and the jumps between them are all the same size. If a term is missing, you can spot the gap right away.

The blue circles are known terms and the yellow circle is the missing term. Each green arc shows a jump of +3. Since 8 + 3 = 11, the missing 3rd term is 11.

Notice how the arcs are all the same size. That equal spacing is what makes a pattern "arithmetic." On the SHSAT, the missing term could be anywhere—at the start, in the middle, or at the end. The strategy is always the same: find the common difference first.

SECTION 4

The Math Behind the Pattern

You only need two small formulas to handle any arithmetic pattern question on the SHSAT. Let's look at each one.

COMMON DIFFERENCE

d = next term − previous term

Pick any two consecutive (side-by-side) terms you know. Subtract the earlier one from the later one. The result is d, the common difference.

NTH TERM FORMULA

aₙ = a₁ + (n − 1) × d

aₙ = the value of the nth term, a₁ = the first term, n = the position number of the term you want, d = common difference.

Most SHSAT problems can be solved just by finding d and then adding or subtracting. The nth-term formula is a shortcut when the missing term is far from the ones you know.

FINDING A MISSING TERM BETWEEN TWO KNOWN TERMS

d = (later known term − earlier known term) ÷ (number of gaps between them)

When the missing term is in the middle, count how many gaps (steps) separate the two known terms. Divide the total change by the number of gaps to get d.

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SHSAT TIP

Always double-check your common difference by testing it with every known term in the sequence. If even one term does not fit, recheck your work!

SECTION 5

Three Scenarios for the Missing Term

On the SHSAT, the missing term could appear at the beginning, in the middle, or at the end of the pattern. Each position requires a slightly different approach, but the underlying idea is the same: find the common difference, then use it.

Each box shows a different place the missing term can hide. In every case, start by finding d using two consecutive known terms, then work toward the blank.

✓ End missing: Add the common difference to the last known term.
✓ Middle missing: Find d from two known neighbors, then add or subtract to reach the gap.
✓ Start missing: Subtract the common difference from the earliest known term.

SECTION 6

Step-by-Step Worked Example

Let's walk through a full problem just like you would see on the SHSAT.

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PROBLEM

Find the missing term in the sequence: 3, 11, ?, 27, 35.

Solving for the Missing 3rd Term

1
Step 1 — Identify what you know

The sequence is 3, 11, ?, 27, 35. The 1st term is 3, the 2nd term is 11, the 4th term is 27, and the 5th term is 35. The 3rd term is missing.
2
Step 2 — Find the common difference using two consecutive known terms

Look at terms 4 and 5, which are both known and sit next to each other: 35 − 27 = 8. So d = 8.

d = 8
3
Step 3 — Verify the common difference

Check with terms 1 and 2: 11 − 3 = 8. Great—it matches!
4
Step 4 — Use d to find the missing term

The missing term is the 3rd term. Start from the 2nd term (11) and add d: 11 + 8 = 19.

Missing term = 19
5
Step 5 — Double-check the full sequence

The complete sequence is now 3, 11, 19, 27, 35. Check every gap: 11 − 3 = 8, 19 − 11 = 8, 27 − 19 = 8, 35 − 27 = 8. Every difference is 8. The answer is confirmed.

✓ Sequence verified: 3, 11, 19, 27, 35

SECTION 7

Common Mistakes & How to Avoid Them

Even strong students can lose easy points on pattern questions by falling into a few common traps. Here is a quick comparison of what to do and what NOT to do.

Mistake	Why It Happens	How to Fix It
Using non-consecutive terms to find d without adjusting	Students subtract any two known terms and call the result d, even when they are not next to each other.	Divide the difference by the number of gaps between those terms to get d.
Adding when you should subtract (or vice versa)	Students always add d, forgetting that the missing term might come before a known term.	If you need a term to the LEFT, subtract d. If to the RIGHT, add d.
Forgetting that d can be negative	Students assume patterns always increase.	A decreasing sequence like 20, 17, 14, 11 has d = −3. Always compute d using subtraction.
Skipping the verification step	Students rush and don't check whether d works for all given terms.	Always plug your answer back in and make sure every gap equals d.

Common arithmetic pattern mistakes on the SHSAT

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KEY TAKEAWAY

Think of finding d like figuring out a bus schedule. If a bus arrives at 2:00 and 2:15, the gap is 15 minutes. But if you only know the 2:00 and the 2:30 arrivals and a bus came in between, you need to divide the 30-minute gap by 2 to get the 15-minute interval.

SECTION 8

Arithmetic Patterns vs. Other Patterns

Arithmetic patterns are just one type of sequence. As you continue in math, you will meet other kinds. Here is a quick comparison so you know what to watch for.

Feature	Arithmetic Pattern	Geometric Pattern
Rule	Add (or subtract) a constant	Multiply (or divide) by a constant
Example	2, 5, 8, 11, 14	3, 6, 12, 24, 48
Key value	Common difference (d)	Common ratio (r)
How to check	Subtract consecutive terms; result is constant	Divide consecutive terms; result is constant
SHSAT frequency	Very common	Less common

Arithmetic vs. Geometric Patterns

For the SHSAT, you will mostly see arithmetic patterns. But if subtracting consecutive terms gives different results each time, check whether dividing them gives a constant ratio instead. That means you are looking at a geometric pattern, and you would multiply or divide to find the missing term.

SECTION 9

Practice Problems

Try these five problems on your own. They start easy and get harder. After each one, read the answer explanation to make sure your reasoning is solid.

Question 1: CONCEPTUAL

In the sequence 4, 9, 14, 19, 24, what is the common difference? How do you know it is an arithmetic pattern?

Show answer

The common difference is 5, because each term is 5 more than the one before it: 9 − 4 = 5, 14 − 9 = 5, 19 − 14 = 5, and 24 − 19 = 5. Since this difference is the same every time, the sequence is arithmetic. If the differences changed from one pair to the next, it would not be arithmetic.

Question 2: BASIC CALCULATION

Find the missing term: 6, 13, 20, ?, 34.

Show answer

First, find d: 13 − 6 = 7. Verify: 20 − 13 = 7. Good. The missing 4th term is 20 + 7 = 27. Check: 34 − 27 = 7. Every gap is 7, so the answer is 27.

Question 3: INTERMEDIATE

Find the missing term: 50, ?, ?, 29, 22. (Two terms are missing, but you only need to find the 2nd term.)

Show answer

Use the two known consecutive terms at the end: 22 − 29 = −7, so d = −7. This is a decreasing pattern. Working backward from 29: the 3rd term is 29 − (−7) = 29 + 7 = 36. Then the 2nd term is 36 + 7 = 43. Verify from the start: 50 − 43 = 7 (absolute value matches). The 2nd term is 43.

Question 4: APPLIED

Maya saves money every week. After Week 1 she has $15. After Week 4 she has $33. If her savings follow an arithmetic pattern, how much did she have after Week 3?

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Week 1 = $15, Week 4 = $33. The number of gaps between Week 1 and Week 4 is 3. So d = (33 − 15) ÷ 3 = 18 ÷ 3 = 6. That means Maya saves $6 each week. Week 2 = 15 + 6 = $21. Week 3 = 21 + 6 = $27. After Week 3, Maya had $27.

Question 5: CRITICAL THINKING

In an arithmetic sequence, the 3rd term is 10 and the 7th term is 34. What is the 5th term?

Show answer

From the 3rd term to the 7th term, there are 7 − 3 = 4 gaps. The total change is 34 − 10 = 24. So d = 24 ÷ 4 = 6. Now use the nth-term idea: the 5th term is 2 gaps after the 3rd term, so it equals 10 + 2 × 6 = 10 + 12 = 22. To verify, the 4th term = 16, 5th = 22, 6th = 28, 7th = 34. All differences are 6. The 5th term is 22.

SUMMARY

Lesson Summary

Summary

An arithmetic pattern is a sequence where each term is formed by adding or subtracting a constant called the common difference (d). To find a missing term, first calculate d by subtracting two consecutive known terms. If the known terms are not next to each other, divide the total change by the number of gaps. Then add or subtract d to reach the missing position.

Remember the nth-term formula aₙ = a₁ + (n − 1) × d for jumping directly to any term. Always verify your answer by plugging it back in and checking that every difference equals d. Watch out for negative common differences in decreasing sequences, and don't confuse arithmetic patterns with geometric patterns (which use multiplication). Master these steps, and missing-term questions on the SHSAT will become some of the easiest points you earn.