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Missing
Sequence Terms

Learn to identify patterns and find missing values in arithmetic, geometric, and other sequences on the ISEE.

ISEE Middle Level

What Is a Sequence?

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

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

Why Sequences Matter on the ISEE

The ISEE Quantitative Reasoning section frequently tests your ability to spot patterns and predict values. These questions appear in two forms:

Find the Next Term
You are given several terms and asked what comes next in the sequence.
Fill the Gap
A term in the middle is missing and you must figure out what belongs there.

Three Main Types of Sequences

Arithmetic
Add or subtract the same number each time. Example: \(3, 7, 11, 15, \ldots\)
Geometric
Multiply or divide by the same number each time. Example: \(2, 6, 18, 54, \ldots\)
Other Patterns
Alternating, squares, two-rule, or repeating patterns. Example: \(1, 4, 9, 16, \ldots\)

The Common Difference

In an arithmetic sequence, the constant value added each time is called the common difference, written as \( d \).

4 10 16 22 ? +6 +6 +6 +6
\( d = 6 \rightarrow \text{missing term} = 22 + 6 = 28 \)

The Common Ratio

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

3 6 ? 24 48
\( \times 2 \)
\( \times 2 \)
\( \times 2 \)
\( \times 2 \)
\( r = 2 \rightarrow \text{missing term} = 6 \times 2 = 12 \)

Arithmetic vs. Geometric

Arithmetic Geometric
Rule Add/subtract a constant \( d \) Multiply/divide by a constant \( r \)
How to Find It \( d = a_{n+1} - a_n \) \( r = a_{n+1} \div a_n \)
Example 5, 9, 13, 17, ... 2, 10, 50, 250, ...
Growth Steady, linear Rapid, exponential
Key Clue Differences are the same Ratios are the same

How to Find a Missing Term

1
List the differences. Subtract each term from the next one to see if you get a constant \( d \).
2
Check for ratios. If differences vary, divide each term by the previous one. Look for a constant \( r \).
3
Look for other patterns. Squares (\(1, 4, 9, 16\)), alternating signs, or two-step rules.
4
Apply the rule forward or backward to fill in the missing term.
5
Verify. Plug the answer back in and confirm the pattern holds for every pair of terms.

Example 1: Arithmetic Sequence

Find the missing term: \( 7, \; 12, \; \boxed{?}, \; 22, \; 27 \)

7 12 ? 22 27 +5 +5 +5 +5
\( ? = 12 + 5 = 17 \)

Example 2: Geometric Sequence

Find the missing term: \( 4, \; 12, \; 36, \; \boxed{?}, \; 324 \)

4 12 36 ? 324
\( \times 3 \)
\( \times 3 \)
\( \times 3 \)
\( \times 3 \)
\( ? = 36 \times 3 = 108 \)

Example 3: Special Pattern

Find the missing term: \( 1, \; 4, \; 9, \; \boxed{?}, \; 25 \)

1 4 9 ? 25 n = 1 n = 2 n = 3 n = 4 n = 5
\( 1^2 \)
\( 2^2 \)
\( 3^2 \)
\( 4^2 \)
\( 5^2 \)
\( ? = 4^2 = 16 \)

Working Backwards

Sometimes the missing term is before the given terms. You reverse the rule:

? 20 27 34 41 +7 +7 +7 -7
\( ? = 20 - 7 = 13 \)

Tricky: Two-Step Patterns

Some ISEE sequences combine two operations or alternate between them:

Alternating Rule
\(2, 6, 4, 12, 10, \ldots\)
Pattern: \( \times 3, -2, \times 3, -2, \ldots \)
Increasing Difference
\(1, 2, 4, 7, 11, \ldots\)
Differences: +1, +2, +3, +4, ...

Common Mistakes to Avoid

✕ Assuming Addition
Always check: if the differences between terms aren't constant, the sequence may be geometric (multiply) or another pattern.
✕ Using Only Two Terms
A single pair of terms can match many rules. Verify the pattern across at least three consecutive pairs before deciding.
✕ Wrong Direction
If the blank is earlier in the sequence, you must reverse the rule (subtract \( d \) or divide by \( r \)), not apply it forward.

Quick-Reference Summary

Step Action Look For
1 Subtract consecutive terms Same difference \( d \)
2 Divide consecutive terms Same ratio \( r \)
3 Compare term to position Squares, cubes, formulas
4 Check differences of differences Increasing/decreasing gap
5 Apply rule and verify Pattern holds for all terms

Question 1 of 5

What is the missing term in the sequence?
\( 3, \; 8, \; 13, \; \boxed{?}, \; 23 \)

Question 2 of 5

Find the missing term:
\( 5, \; 15, \; 45, \; \boxed{?}, \; 405 \)

Question 3 of 5

What number is missing?
\( \boxed{?}, \; 10, \; 16, \; 22, \; 28 \)

Question 4 of 5

Find the missing term in this sequence:
\( 1, \; 3, \; 6, \; 10, \; \boxed{?} \)

Question 5 of 5

What is the missing term?
\( 2, \; 6, \; 18, \; 54, \; \boxed{?} \)