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Missing
Sequence Terms

Master the art of identifying patterns and finding hidden terms in arithmetic, geometric, and other sequences.

SSAT Upper Level: Quantitative

What Is a Sequence?

A sequence is an ordered list of numbers that follows a specific pattern or rule. Each number in the list is called a term.

2 Term 1 +3 5 Term 2 +3 8 Term 3 +3 11 Term 4 +3 ?

Key Vocabulary

Term
Each individual number in the sequence. Often written as \( a_1, a_2, a_3, \dots \)
Common Difference
The constant value added between consecutive terms of an arithmetic sequence: \( d = a_{n+1} - a_n \)
Common Ratio
The constant value multiplied between consecutive terms of a geometric sequence: \( r = \frac{a_{n+1}}{a_n} \)

Why Sequences Matter on the SSAT

Sequence questions appear consistently on the SSAT Upper Level quantitative section. They test your ability to recognize patterns, apply rules, and compute efficiently. A strong grip on sequences can earn you quick, confident points.

Pattern Recognition
Spotting the rule between terms quickly is the core skill.
Formula Application
Using \( a_n = a_1 + (n-1)d \) or \( a_n = a_1 \cdot r^{n-1} \) saves time.
Speed and Accuracy
Once you identify the type, the answer follows in seconds.

Arithmetic Sequences: The Rule

In an arithmetic sequence, each term is found by adding a constant called the common difference \( d \) to the previous term.

4 \( a_1 \) 10 \( a_2 \) 16 \( a_3 \) 22 \( a_4 \) +6 +6 +6 \( d = 6 \)

Geometric Sequences: The Rule

In a geometric sequence, each term is found by multiplying the previous term by a constant called the common ratio \( r \).

3 \( a_1 \) 6 \( a_2 \) 12 \( a_3 \) 24 \( a_4 \) ×2 ×2 ×2 \( r = 2 \)

Arithmetic vs. Geometric

Arithmetic Geometric
Operation Add (or subtract) Multiply (or divide)
Constant \( d \) (common difference) \( r \) (common ratio)
Formula \( a_n = a_1 + (n-1)d \) \( a_n = a_1 \cdot r^{n-1} \)
Test Differences are equal Ratios are equal
Example 5, 8, 11, 14, ... 5, 15, 45, 135, ...

How to Find the Missing Term

1
List the known terms and note their positions (\( a_1, a_2, a_3, \dots \)).
2
Compute differences: subtract each term from the next. If they are constant, you have \( d \).
3
Compute ratios: if differences vary, divide each term by the previous one. If constant, you have \( r \).
4
Apply the rule forward or backward to fill in the missing term(s).
5
Verify: plug the answer back in and confirm the pattern holds for all terms.

Example 1: Arithmetic

Problem: Find the missing term: 7, 13, ?, 25, 31

7 13 ? 25 31 +6 +6

Example 2: Geometric

Problem: Find the missing term: 2, 6, 18, ?, 162

2 6 18 ? 162 ×3 ×3

Example 3: Other Patterns

Problem: Find the missing term: 1, 4, 9, ?, 25

Beyond Basic Patterns

Second Differences
If first differences aren't constant, check the differences of the differences. Constant second differences indicate a quadratic pattern.
Alternating
Some sequences alternate between two rules: e.g., add 2, then multiply by 3, then add 2, then multiply by 3...
Fibonacci-Style
Each term is the sum of the two preceding terms: \( a_n = a_{n-1} + a_{n-2} \).

Working Backwards

Sometimes the missing term is before the known terms. You reverse the operation:

1
Arithmetic: subtract \( d \) instead of adding. If \( a_3 = 20 \) and \( d = 4 \), then \( a_2 = 20 - 4 = 16 \).
2
Geometric: divide by \( r \) instead of multiplying. If \( a_3 = 36 \) and \( r = 3 \), then \( a_2 = \frac{36}{3} = 12 \).
3
General: Identify which operation produces the next term, then undo it to go backwards.

Handling Gaps Between Terms

When terms are not consecutive, use the formula to bridge the gap.

Common Mistakes

✕ Assuming Arithmetic
Seeing "the numbers go up" and immediately adding. Always check: are the differences actually constant?
✕ Wrong Gap Count
The number of steps between term \( a_m \) and term \( a_n \) is always the difference in their indices: \( n - m \). For example, from \( a_2 \) to \( a_5 \) is \( 5 - 2 = 3 \) steps.
✕ Sign Errors
Sequences can decrease: \( d \) can be negative (e.g., 20, 15, 10 has \( d = -5 \)). Don't forget the minus sign.

Summary: Your Toolkit

Step What to Do Why
1 Subtract consecutive terms Check for constant \( d \)
2 Divide consecutive terms Check for constant \( r \)
3 Check second differences Detect quadratic patterns
4 Look for powers/special patterns Squares, cubes, Fibonacci
5 Apply rule and verify Confirm answer fits all terms

Question 1 of 5

What is the missing term in the sequence 3, 8, 13, ?, 23?

Question 2 of 5

What is the missing term in the sequence 4, 12, 36, ?, 324?

Question 3 of 5

In an arithmetic sequence, \( a_3 = 14 \) and \( a_7 = 30 \). What is \( a_5 \)?

Question 4 of 5

What is the missing term in the sequence 1, 1, 2, 3, 5, ?, 13?

Question 5 of 5

What is the missing first term in the sequence ?, 10, 20, 40, 80?