01 / 20
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Predicting
Later Terms

Learn to spot patterns in sequences and use them to figure out any term, no matter how far along it is.

SSAT Middle Level: Quantitative

What Is a Sequence?

A sequence is an ordered list of numbers that follows a rule. Each number in the list is called a term. The position of a term is its term number.

3 Term 1 6 Term 2 9 Term 3 12 Term 4 ? Term 5
\(+3\)
\(+3\)
\(+3\)
\(+3\)

Why Predict Later Terms?

The SSAT often gives you the first few terms and asks for a term far down the list: the 10th term, the 20th term, or even the 100th term. You need a strategy that is faster than listing every single term.

Counting One-by-One
Works for small positions but is slow and error-prone for the 50th or 100th term.
Using a Formula
Jump directly to any term once you know the pattern: fast, reliable, and exactly what the test wants.

Common Pattern Types

Add / Subtract
Same number added or subtracted each step. Example: 5, 8, 11, 14, ...
Multiply / Divide
Same number multiplied or divided each step. Example: 2, 6, 18, 54, ...
Repeating Cycle
A group of terms repeats over and over. Example: 1, 4, 7, 1, 4, 7, ...

The Add/Subtract Rule

When each term is found by adding (or subtracting) the same number, the sequence is called arithmetic. The constant difference is called \( d \).

Seeing the Formula in Action

5
\(n=1\)
 8
\(n=2\)
 11
\(n=3\)
 14
\(n=4\)
 17
\(n=5\)
\(+3\)
\(+3\)
\(+3\)
\(+3\)

To reach Term 5:

Start at 5, add 3 exactly \( (5 - 1) = 4 \) times

\( 5 + 4 \times 3 = 5 + 12 = 17 \)

Multiply / Divide Patterns

When each term is found by multiplying (or dividing) by the same number, we call the constant the common ratio \( r \).

2 Term 1 6 Term 2 18 Term 3 54 Term 4 ? Term 5
\(\times 3\)
\(\times 3\)
\(\times 3\)
\(\times 3\)

Repeating Cycle Patterns

Some sequences repeat a fixed group of terms over and over. To find a later term, figure out where it falls in the cycle using remainders.

1 4 7 1 4 7 1 ? Cycle length = 3 | What is the 8th term?
\( 8 \div 3 = 2 \) remainder \( 2 \) → 2nd item in cycle = \( 4 \)

Add vs Multiply Patterns

Add / Subtract Multiply / Divide
How to spot it Differences between terms are constant Ratios between terms are constant
Example 4, 7, 10, 13, ... 3, 12, 48, 192, ...
Growth speed Grows steadily Grows very fast
Formula \( a_1 + (n-1)d \) \( a_1 \times r^{n-1} \)
SSAT frequency Very common Less common

Step-by-Step Strategy

1
Find the differences (or ratios). Subtract each pair of consecutive terms. If the differences are the same, it is an add/subtract pattern.
2
Identify \( a_1 \) and \( d \) (or \( r \)). Write down the first term and the common difference (or ratio).
3
Plug into the formula. Use \( a_n = a_1 + (n-1) \times d \) for add/subtract, or \( a_n = a_1 \times r^{n-1} \) for multiply.
4
Compute and check. Do the arithmetic carefully. Verify your answer makes sense (bigger or smaller than nearby terms?).

Example 1: Find the 15th Term

Sequence: 7, 11, 15, 19, 23, ... What is the 15th term?

7 11 15 19 23
\(+4\)
\(+4\)
\(+4\)
\(+4\)
1
Differences: \( 11 - 7 = 4 \), \( 15 - 11 = 4 \). Common difference \( d = \) \(4\).
2
First term: \( a_1 = \) \(7\). We want \( n = 15 \).
3
\( a_{15} = 7 + (15 - 1) \times 4 = 7 + 56 = 63 \).

Example 2: Multiply Pattern

Sequence: 5, 10, 20, 40, ... What is the 7th term?

1
Check differences: \( 10 - 5 = 5 \), \( 20 - 10 = 10 \). Not constant. Check ratios: \( 10 \div 5 = 2 \), \( 20 \div 10 = 2 \). Common ratio \( r = \) \(2\).
2
\( a_1 = 5 \), \( r = 2 \), \( n = 7 \).
3
\( a_7 = 5 \times 2^{6} = 5 \times 64 = 320 \).

Example 3: Repeating Cycle

Sequence: 2, 5, 8, 2, 5, 8, 2, 5, 8, ... What is the 20th term?

1
The cycle is \(2, 5, 8\) and repeats every \(3\) terms.
2
Divide: \( 20 \div 3 = 6 \) remainder \(2\).
3
Remainder \(2\) means the 20th term is the 2nd item in the cycle: \(5\).

Common Mistakes

✕ Using \( n \) instead of \( n - 1 \)
The formula uses \( n - 1 \) jumps, not \( n \). Adding one extra jump gives an answer that is too large by \( d \).
✕ Mixing up add and multiply
Always check: are the differences constant or the ratios? Using the wrong formula gives a wildly wrong answer.
✕ Remainder 0 in cycles
If the remainder is 0, the answer is the last item in the cycle, not the first. Many students pick the wrong one.

Quick Reference Summary

Pattern Type How to Identify Formula / Method
Add / Subtract Constant difference \( d \) \( a_1 + (n-1)d \)
Multiply / Divide Constant ratio \( r \) \( a_1 \times r^{n-1} \)
Repeating Cycle Block of terms repeats Divide \( n \) by cycle length, use remainder

Question 1 of 5

The sequence is 4, 9, 14, 19, 24, ... What is the 20th term?

Question 2 of 5

The sequence is 3, 6, 12, 24, ... What is the 6th term?

Question 3 of 5

The pattern A, B, C, D repeats: A, B, C, D, A, B, C, D, ... What is the 25th letter?

Question 4 of 5

The sequence is 100, 93, 86, 79, ... What is the 12th term?

Question 5 of 5

The sequence is 1, 3, 9, 27, 81, ... What is the 8th term?