01 / 20

Missing Terms in
Sequences

Learn to identify patterns, apply sequence formulas, and determine unknown terms in arithmetic, geometric, and other sequences.

ISEE Upper Level Quantitative Reasoning

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.

3 Term 1 +5 8 Term 2 +5 13 Term 3 +5 ? Term 4

Key Vocabulary

Term
Each individual number in a sequence, denoted \( a_n \) for the \( n \)-th term.
Common Difference
The constant \( d \) added each step in an arithmetic sequence.
Common Ratio
The constant \( r \) multiplied each step in a geometric sequence.

Two Main Types of Sequences

Arithmetic
Each term is formed by adding a constant. Example: \( 2, 5, 8, 11, \ldots \) with \( d = 3 \).
Geometric
Each term is formed by multiplying a constant. Example: \( 4, 12, 36, 108, \ldots \) with \( r = 3 \).

Recognizing which type a sequence is determines which formula you use to find the missing term.

Arithmetic Sequence Formula

5
\( a_1 \)
 9
\( a_2 \)
 13
\( a_3 \)
 17
\( a_4 \)
 +4 +4 +4
\( d = 4 \), so \( a_n = 5 + (n-1) \cdot 4 = 4n + 1 \)

Geometric Sequence Formula

2
\( a_1 \)
 6
\( a_2 \)
 18
\( a_3 \)
 54
\( a_4 \)
 x3 x3 x3
\( r = 3 \), so \( a_n = 2 \cdot 3^{n-1} \)

Arithmetic vs. Geometric

Arithmetic Geometric
Operation Add / subtract Multiply / divide
Constant Common difference \( d \) Common ratio \( r \)
Formula \( a_1 + (n-1)d \) \( a_1 \cdot r^{n-1} \)
Growth Linear (steady) Exponential (fast)
Quick test Differences constant? Ratios constant?

Beyond the Basics: Other Patterns

Perfect Squares
\( 1, 4, 9, 16, 25, \ldots \)
Rule: \( a_n = n^2 \)
Fibonacci-type
\( 1, 1, 2, 3, 5, 8, \ldots \)
Rule: \( a_n = a_{n-1} + a_{n-2} \)
Alternating
\( 3, -6, 12, -24, \ldots \)
Rule: multiply by \( -2 \)

Step-by-Step Strategy

1
List the differences between consecutive terms. If constant, the sequence is arithmetic.
2
List the ratios of consecutive terms. If constant, the sequence is geometric.
3
If neither works, check for squares, cubes, alternating signs, or sums of prior terms.
4
Apply the rule to compute the missing term using the correct formula.
5
Verify by checking that your answer fits the pattern in both directions.

Example 1: Arithmetic

Find the missing term: \( 7, \; 13, \; 19, \; \mathord{?}, \; 31 \)

7 +6 13 +6 19 +6 25 +6 31
\( d = 6 \) → missing term = \( 19 + 6 = 25 \)

Example 2: Geometric

Find the missing term: \( 5, \; 15, \; \mathord{?}, \; 135, \; 405 \)

5 x3 15 x3 45 x3 135 x3 405
\( r = 3 \) → missing term = \( 15 \times 3 = 45 \)

Example 3: Non-Standard Pattern

Find the missing term: \( 1, \; 4, \; 9, \; 16, \; \mathord{?}, \; 36 \)

1
\(1^2\)
 4
\(2^2\)
 9
\(3^2\)
 16
\(4^2\)
 ?
\(5^2\)
 36
\(6^2\)
 +3 +5 +7 +9 +11 Differences grow by 2 each time (odd numbers). Second differences are constant!
Missing term = \( 5^2 = 25 \)

Finding \( d \) or \( r \) from Non-Adjacent Terms

Sometimes the ISEE gives you terms that are not next to each other. Use the formulas to work backward.

Arithmetic
If \( a_m \) and \( a_n \) are known:
$$ d = \frac{a_n - a_m}{n - m} $$
Geometric
If \( a_m \) and \( a_n \) are known:
$$ r = \left(\frac{a_n}{a_m}\right)^{\!\frac{1}{n-m}} $$

Example 4: Working Backwards

In a geometric sequence, \( a_1 = 4 \) and \( a_4 = 108 \). Find \( a_3 \).

1
Set up the formula: \( a_4 = a_1 \cdot r^3 \), so \( 108 = 4 \cdot r^3 \).
2
Solve for \( r \): \( r^3 = \frac{108}{4} = 27 \), so \( r = 3 \).
3
Find \( a_3 \): \( a_3 = 4 \cdot 3^{3-1} = 4 \cdot 9 = \) 36.
4
Verify: \( 4, \; 12, \; 36, \; 108 \) : each ratio is 3. ✓

Common Mistakes

✕ Mixing up \( d \) and \( r \)
Adding when you should multiply, or vice versa. Always check: is the pattern additive or multiplicative?
✕ Off-by-one errors
Using \( n \) instead of \( n - 1 \) in the exponent. Remember: \( a_n = a_1 \cdot r^{n-1} \), not \( r^n \).
✕ Assuming arithmetic
Jumping to "add the difference" without checking. Always compute at least two differences or two ratios first.

Quick Reference Summary

Pattern Type How to Identify Formula
Arithmetic Constant differences \( a_1 + (n-1)d \)
Geometric Constant ratios \( a_1 \cdot r^{n-1} \)
Quadratic Constant 2nd differences \( an^2 + bn + c \)
Fibonacci-type Each term = sum of two prior \( a_{n-1} + a_{n-2} \)
Alternating Signs flip each term Multiply by negative \( r \)

Question 1 of 5

What is the missing term in the sequence \( 3, \; 10, \; 17, \; \mathord{?}, \; 31 \)?

Question 2 of 5

Find the missing term: \( 2, \; 6, \; 18, \; 54, \; \mathord{?} \)

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 next term in the sequence \( 1, \; 1, \; 2, \; 3, \; 5, \; 8, \; \mathord{?} \)?

Question 5 of 5

Find the missing term: \( 1, \; 8, \; 27, \; \mathord{?}, \; 125 \)