Patterns and sequences are one of the most reliably tested topics on the Gauss math contest — appearing in almost every past paper across both the Grade 7 and Grade 8 versions. Whether a question involves a growing pattern where each term increases by a fixed amount, a shrinking pattern where each term decreases, or a repeating pattern that cycles through the same values, the underlying skill is the same: identify the rule, express it algebraically, and use it to find any term in the sequence without counting through every step. This guide covers growing patterns, shrinking patterns, repeating patterns, and pattern rules with full worked examples and Gauss-style practice problems.
Competing or thinking about competing in the Gauss contest? Find everything you need to know here at: Gauss Contest: What Canadian Students Need to Know Before They Compete.
Types of patterns in mathematics
Before working through practice problems it is worth being clear about the different types of patterns that appear in Gauss contest questions, because each type requires a slightly different approach.
| Pattern type | What it does | Example |
|---|---|---|
| Growing pattern | Each term is larger than the previous | 3, 7, 11, 15, 19… |
| Shrinking pattern | Each term is smaller than the previous | 50, 44, 38, 32, 26… |
| Repeating pattern | A fixed set of elements cycles repeatedly | Red, Blue, Green, Red, Blue, Green… |
| Geometric pattern | Each term is multiplied by a fixed ratio | 2, 6, 18, 54, 162… |
The Gauss contest tests all four types. Growing patterns and repeating patterns appear most frequently. Understanding how to find the pattern rule for each type is the core skill this guide develops.
What is a pattern rule?
A pattern rule is a mathematical instruction that describes how to generate every term in a sequence. There are two types of pattern rule worth knowing: the recursive rule and the explicit rule.
Recursive pattern rule
A recursive rule tells you how to get from one term to the next. It describes the operation applied to each term to produce the following term.
For the sequence 3, 7, 11, 15, 19: the recursive rule is “add 4 to each term to get the next term.”
Recursive rules are easy to identify by looking at the differences between consecutive terms. They are useful for generating the next few terms of a sequence but impractical for finding a term far down the sequence — you would have to calculate every term in between.
Explicit pattern rule
An explicit rule — also called the nth term formula — gives the value of any term directly from its position in the sequence without needing to know the previous term.
For the same sequence 3, 7, 11, 15, 19: the explicit rule is “the nth term equals 4n – 1.”
Check: n=1 gives 4(1) – 1 = 3 ✓. n=3 gives 4(3) – 1 = 11 ✓. n=50 gives 4(50) – 1 = 199, without needing to calculate all 50 terms.
The explicit rule is what Gauss contest questions almost always require, because questions typically ask for the value of a term far into the sequence — the 50th term, the 100th term, the 1000th — where counting through the recursive rule is not practical.
How to find the pattern rule for an arithmetic sequence
An arithmetic sequence is any sequence where the difference between consecutive terms is constant. Both growing patterns with constant increase and shrinking patterns with constant decrease are arithmetic sequences.
The explicit rule (nth term formula) for an arithmetic sequence is:
nth term = first term + (n – 1) x common difference
Where the common difference is positive for growing patterns and negative for shrinking patterns.
This can always be simplified to the form nth term = an + b where a is the common difference and b is a constant found by substituting n = 1.
Definition, examples and worksheets
A growing pattern is any sequence where each term is larger than the previous one. The simplest growing patterns are arithmetic — the amount added each time is constant. More complex growing patterns are geometric or follow a more complex rule.
Arithmetic
The sequence 5, 9, 13, 17, 21 is a growing pattern. Each term increases by 4. The common difference is 4.
Finding the nth term: First term = 5.
Common difference = 4.
nth term = 5 + (n-1) x 4 = 5 + 4n – 4 = 4n + 1.
Check: n=1: 4+1 = 5 ✓.
n=4: 16+1 = 17 ✓.
Worksheets — practice
Problem 1
A growing pattern begins 8, 13, 18, 23, 28…
Find the pattern rule and the 40th term.
Solution: Common difference = 13 – 8 = 5. nth term = 8 + (n-1) x 5 = 8 + 5n – 5 = 5n + 3.
40th term: 5(40) + 3 = 200 + 3 = 203.
Answer: nth term = 5n + 3, 40th term = 203
Problem 2
A growing pattern has first term 6 and common difference 7. Which term in the pattern equals 97?
Solution: nth term = 6 + (n-1) x 7 = 6 + 7n – 7 = 7n – 1.
Set equal to 97: 7n – 1 = 97, 7n = 98, n = 14.
Answer: The 14th term equals 97
Problem 3 — Gauss style growing pattern
A pattern of squares is arranged in rows. Row 1 has 3 squares, Row 2 has 7 squares, Row 3 has 11 squares. If this pattern continues, how many squares are in Row 25?
Solution: Sequence: 3, 7, 11… Common difference = 4. nth term = 3 + (n-1) x 4 = 3 + 4n – 4 = 4n – 1.
Row 25: 4(25) – 1 = 100 – 1 = 99.
Answer: 99 squares
Shrinking patterns — definition and examples
A shrinking pattern is a sequence where each term is smaller than the previous one. The simplest shrinking patterns have a constant decrease — a negative common difference.
What is a shrinking pattern?
A shrinking pattern decreases by the same amount each time. The common difference is negative.
The sequence 100, 93, 86, 79, 72 is a shrinking pattern. Each term decreases by 7. The common difference is -7.
Finding the nth term: First term = 100. Common difference = -7. nth term = 100 + (n-1) x (-7) = 100 – 7n + 7 = 107 – 7n.
Check: n=1: 107-7 = 100 ✓. n=3: 107-21 = 86 ✓.
Shrinking pattern practice
Problem 4
A shrinking pattern begins 85, 79, 73, 67, 61…
Find the nth term formula and determine which term first becomes negative.
Solution: Common difference = 79 – 85 = -6. nth term = 85 + (n-1)(-6) = 85 – 6n + 6 = 91 – 6n.
Set less than 0: 91 – 6n < 0, 91 < 6n, n > 15.17.
The 16th term is the first negative term. Check: 16th term = 91 – 6(16) = 91 – 96 = -5. Negative ✓. 15th term = 91 – 6(15) = 91 – 90 = 1. Still positive ✓.
Answer: nth term = 91 – 6n, the 16th term is the first negative term
Problem 5 — Gauss style shrinking pattern
A shrinking pattern starts at 200 and decreases by 11 each step. What is the last positive term in the sequence?
Solution: nth term = 200 + (n-1)(-11) = 200 – 11n + 11 = 211 – 11n.
Set greater than 0: 211 – 11n > 0, n < 19.18.
The last positive term is the 19th term. 19th term = 211 – 11(19) = 211 – 209 = 2.
Answer: 2
Increasing patterns — when patterns grow in different ways
Not all increasing patterns have a constant difference. Some increasing patterns have differences that themselves follow a pattern — called second differences. These are slightly more complex and appear on Gauss Part B and C questions.
Identifying second differences
For the sequence 1, 4, 9, 16, 25 (the square numbers), the first differences are 3, 5, 7, 9 — increasing, not constant. The second differences (differences of the differences) are 2, 2, 2, 2 — constant.
When second differences are constant, the nth term follows a quadratic formula (involves n²). For the square numbers, the nth term is simply n².
Example — increasing pattern with second differences
Find the nth term for the sequence 2, 5, 10, 17, 26…
First differences: 3, 5, 7, 9 — increasing by 2 each time. Second differences: 2, 2, 2 — constant.
Since second differences are constant at 2, the nth term contains n². Start with n²: 1, 4, 9, 16, 25.
Compare to sequence: 2-1=1, 5-4=1, 10-9=1, 17-16=1, 26-25=1. The difference is always 1.
nth term = n² + 1.
Check: n=1: 1+1=2 ✓. n=4: 16+1=17 ✓.
Answer: nth term = n² + 1
For students preparing for the Gauss, it is worth being aware that quadratic sequences exist and that the second difference method identifies them. At Grade 7 and 8 level, most Gauss pattern questions use arithmetic (linear) sequences, but second difference problems appear occasionally on harder Part C questions.
Repeating patterns — definition, examples and cycle problems
A repeating pattern is a sequence where a fixed set of elements — the core — repeats over and over. Unlike growing or shrinking patterns, repeating patterns do not increase or decrease — they cycle.
What is a repeating pattern?
A repeating pattern has a core that cycles indefinitely. The core is the smallest group of elements that repeats.
For the sequence: Red, Blue, Green, Red, Blue, Green, Red, Blue, Green…
The core is Red, Blue, Green. Its length is 3. The pattern repeats every 3 elements.
Finding any term in a repeating pattern
The key tool for repeating patterns is division with remainder. To find the nth element:
- Find the length of the repeating core
- Divide n by the core length
- The remainder tells you which position in the core the nth element occupies
- If the remainder is 0, the element is the last in the core
Repeating pattern example
A repeating pattern follows: A, B, C, D, A, B, C, D…
What is the 53rd element?
Core length = 4. 53 ÷ 4 = 13 remainder 1.
Remainder 1 corresponds to position 1 in the core, which is A.
Answer: A
Repeating patterns — harder example
The letters of the word GAUSS are repeated in order: G, A, U, S, S, G, A, U, S, S…
What is the 74th letter?
Core: G, A, U, S, S. Core length = 5. 74 ÷ 5 = 14 remainder 4.
Remainder 4 corresponds to position 4 in the core, which is S.
Answer: S
Key insight: The remainder of dividing the position by the core length always gives the position within the core. A remainder of 0 means the element is the last in the core — be careful not to say the element is “position 0.”
Repeating patterns — Gauss contest practice
Problem 6 — basic repeating pattern
A sequence of shapes repeats: circle, triangle, square, circle, triangle, square…
What is the 100th shape?
Solution: Core: circle, triangle, square. Core length = 3. 100 ÷ 3 = 33 remainder 1.
Remainder 1 = position 1 = circle.
Answer: Circle
Problem 7 — repeating pattern with numbers
A sequence repeats: 1, 3, 5, 2, 1, 3, 5, 2…
What is the sum of the first 50 terms?
Solution: Core: 1, 3, 5, 2. Core length = 4. Core sum = 1+3+5+2 = 11.
50 ÷ 4 = 12 remainder 2.
12 complete cycles contribute: 12 x 11 = 132. Remaining 2 terms are the first 2 of the core: 1 + 3 = 4.
Total sum = 132 + 4 = 136.
Answer: 136
Key insight: For sum problems with repeating patterns, find the sum of one complete core, multiply by the number of complete cores, then add the partial core at the end. This structure works for any repeating pattern sum problem.
Problem 8 — Gauss style repeating pattern
The digits 1, 2, 3, 4, 5 are written repeatedly: 1, 2, 3, 4, 5, 1, 2, 3, 4, 5…
What is the sum of the first 73 digits?
Solution: Core: 1, 2, 3, 4, 5. Core length = 5. Core sum = 15.
73 ÷ 5 = 14 remainder 3.
14 complete cycles: 14 x 15 = 210. Remaining 3 digits: 1 + 2 + 3 = 6.
Total = 210 + 6 = 216.
Answer: 216
Problem 9 — days of the week repeating pattern
January 1st is a Wednesday. What day of the week is March 15th of the same year? (January has 31 days, February has 28 days in a non-leap year.)
Solution: Days from January 1st to March 15th: January: 30 remaining days (Jan 2 to Jan 31). February: 28 days. March: 15 days (March 1 to March 15). Total days after January 1st: 30 + 28 + 15 = 73 days.
Days of the week repeat with core length 7. 73 ÷ 7 = 10 remainder 3.
Wednesday is position 1. Position 1 + 3 = position 4 from Wednesday. Wednesday → Thursday (1) → Friday (2) → Saturday (3).
Answer: Saturday
Key insight: Calendar problems are repeating pattern problems with core length 7. Count the total number of days from the known date to the target date, divide by 7, and use the remainder to count forward from the known day.
Pattern worksheets — mixed practice
These problems mix all four pattern types and are written in the style of Gauss contest questions.
Problem 10
A growing pattern begins 4, 11, 18, 25… and a shrinking pattern begins 100, 94, 88, 82…
At what position do the two sequences first have the same value?
Solution: Growing: nth term = 4 + (n-1)(7) = 7n – 3. Shrinking: nth term = 100 + (n-1)(-6) = 106 – 6n.
Set equal: 7n – 3 = 106 – 6n. 13n = 109. n = 109/13 — not a whole number.
The sequences never share a value at the same position.
To find if they share any common value, note that growing sequence values are 4, 11, 18, 25… (all ≡ 4 mod 7) and shrinking sequence values are 100, 94, 88… (all ≡ 4 mod 6).
Common values must satisfy both: equal to 4 mod 7 and equal to 4 mod 6. 4, 11, 18, 25, 46, 88… checking which appear in both: Growing: 4, 11, 18, 25, 32, 39, 46, 53, 60, 67, 74, 81, 88… Shrinking: 100, 94, 88… — 88 appears in both.
Answer: 88 is the first value that appears in both sequences
Problem 11
A pattern of tiles is shown. In Figure 1 there are 5 tiles, in Figure 2 there are 9 tiles, in Figure 3 there are 13 tiles. How many tiles are needed in total for the first 10 figures?
Solution: nth term = 5 + (n-1)(4) = 4n + 1.
Sum of first 10 terms: Using the arithmetic series sum formula: S = n/2 x (first term + last term).
10th term = 4(10) + 1 = 41. S = 10/2 x (5 + 41) = 5 x 46 = 230.
Answer: 230 tiles
Key insight: The sum of an arithmetic sequence is n/2 x (first term + last term). This formula avoids adding all terms individually and is worth memorising for Gauss questions that ask for a total rather than a single term. For more on sequences and finding terms, our grade 8 algebra worksheets guide covers the algebraic approach to patterns in depth — see Grade 8 Algebra Worksheets: Practice Problems for the Gauss Contest for worked examples and practice.
Problem 12 — hardest
A sequence has the property that each term is the sum of the two terms before it. The first term is 1 and the second term is 3. What is the 10th term?
Solution: This is a Fibonacci-style sequence — each term is the sum of the two before it.
Term 1: 1
Term 2: 3
Term 3: 1 + 3 = 4
Term 4: 3 + 4 = 7
Term 5: 4 + 7 = 11
Term 6: 7 + 11 = 18
Term 7: 11 + 18 = 29
Term 8: 18 + 29 = 47
Term 9: 29 + 47 = 76
Term 10: 47 + 76 = 123
Answer: 123
Key insight: Some Gauss sequences are defined recursively and have no simple explicit formula — you must generate the terms one by one. Recognising “each term is the sum of the two before” is the Fibonacci-style rule and generates the sequence correctly. Work carefully and check each term as you go.
Pattern rules — quick reference
| Sequence type | How to identify | nth term formula | Example |
|---|---|---|---|
| Arithmetic growing | Constant positive difference | first + (n-1)d | 3, 7, 11: nth = 4n – 1 |
| Arithmetic shrinking | Constant negative difference | first + (n-1)d (d negative) | 20, 17, 14: nth = 23 – 3n |
| Geometric | Constant ratio between terms | first x r^(n-1) | 2, 6, 18: nth = 2 x 3^(n-1) |
| Repeating | Fixed core cycles | Use division with remainder | A,B,C,A,B,C: use n mod 3 |
| Quadratic | Constant second differences | Contains n² | 1, 4, 9, 16: nth = n² |
How to use these pattern worksheets
These pattern worksheets are most effective when worked through in order — growing and shrinking patterns first, then repeating patterns, then mixed problems. The skills build on each other and the mixed problems at the end require all of them simultaneously.
For every problem, write the pattern rule explicitly before trying to find any specific term. Students who try to find the 50th term by counting forward from the 10th term make significantly more errors than students who find the explicit formula first and substitute n = 50 directly.
When working through growing patterns worksheets problems, always verify the formula by checking it against at least two known terms before using it to find an unknown term. A formula that works for n=1 but fails for n=2 has been set up incorrectly.
For repeating patterns, always write out the core explicitly and count its length before dividing. Miscounting the core length is the most common error on these problems and it is completely preventable by being systematic.
These pattern worksheets connect directly to other Gauss topics. Rate problems often involve arithmetic sequences — if you found the unit rate guide useful, the same algebraic reasoning applies to growing patterns. For building the fraction skills needed to work with geometric sequences, our guide on equivalent fractions covers the proportional reasoning in depth — see Equivalent Fractions Worksheet: Practice Problems and Examples for practice.
And for the algebra needed to find and use nth term formulas, our complete set of algebra practice and worked examples covers everything from expressions through to word problems — see Grade 8 Algebra Worksheets: Practice Problems for the Gauss Contest.
Frequently Asked Questions
What is a growing pattern? A growing pattern is a sequence where each term is larger than the previous one. The simplest growing patterns are arithmetic — each term increases by the same constant amount called the common difference. The nth term of an arithmetic growing pattern is found using the formula: first term + (n-1) x common difference.
What is a shrinking pattern? A shrinking pattern is a sequence where each term is smaller than the previous one. Like growing patterns, the simplest shrinking patterns are arithmetic with a constant negative common difference. The same nth term formula applies — first term + (n-1) x common difference — where the common difference is a negative number.
What is a repeating pattern? A repeating pattern is a sequence where a fixed group of elements — called the core — cycles repeatedly. To find any term in a repeating pattern, divide the term’s position by the core length and use the remainder to identify which element of the core that position corresponds to.
What is a pattern rule? A pattern rule describes how a sequence is generated. A recursive rule tells you how to get from one term to the next. An explicit rule — the nth term formula — gives the value of any term directly from its position without needing to calculate previous terms. For Gauss contest questions the explicit rule is almost always what is needed.
How do growing patterns appear on the Gauss math contest? Growing patterns appear on the Gauss in several forms: finding the nth term of an arithmetic sequence, identifying how many terms satisfy a given condition, calculating the sum of the first n terms, and pattern problems presented with geometric dot or tile figures where the sequence of term sizes must be identified from the visual. Repeating patterns appear most often in position problems — what is the 100th element — and sum problems.
How do you find the sum of an arithmetic sequence? The sum of the first n terms of an arithmetic sequence is n/2 x (first term + last term). Find the last term using the nth term formula, then apply the sum formula. This is significantly faster than adding all terms individually and is the expected approach on Gauss Part B and C sum problems.
What is the difference between increasing patterns and growing patterns? The terms are often used interchangeably. Strictly, a growing pattern or increasing pattern is any sequence that increases — arithmetic (constant difference), geometric (constant ratio), or more complex. In the context of Gauss contest preparation, growing patterns and increasing patterns both refer to sequences where each term is larger than the previous, with arithmetic sequences being the most commonly tested type.
