Sequences & Patterns Practice Quiz with a Step-by-Step Interactive Lesson
Use the quiz at the top of the page to practice number sequences and patterns: find the next term, identify the sequence rule, and write an \(n\)th term formula. This lesson focuses on the most common pattern types used in schools and exams: arithmetic sequences (constant difference), geometric sequences (constant ratio), recursive sequences, and classic patterns like the Fibonacci sequence, square numbers, cube numbers, triangular numbers, and the prime number sequence. If you want a refresher, click Start lesson to open a step-by-step guide with worked examples.
How this sequences and patterns practice works
1. Take the quiz: answer the sequence questions at the top of the page.
2. Open the lesson (optional): learn reliable strategies (differences, ratios, and formulas) with worked examples.
3. Retry: return to the quiz and apply the pattern rules immediately.
What you will learn in the sequences and patterns lesson
Foundations & vocabulary
Sequence, term, index (e.g., \(a_1, a_2, a_3,\dots\))
Explicit rule (direct \(a_n\)) vs recursive rule (build from previous terms)
Pattern checks: does your rule match every given term?
Arithmetic sequences
Constant difference: \(a_{n}-a_{n-1}=d\)
\(n\)th term formula: \(a_n=a_1+(n-1)d\)
Common exam tasks: next term, \(n\)th term, and "which term equals ...?"
Geometric sequences
Constant ratio: \(\dfrac{a_n}{a_{n-1}}=r\) (when terms are nonzero)
\(n\)th term formula: \(a_n=a_1\cdot r^{\,n-1}\)
Growth patterns: doubling, tripling, and repeated multiplication
Pattern strategies & classic sequences
Difference tables (including second differences for "square-like" patterns)
Fibonacci-style rules: each term is the sum of the previous two
Special sequences: squares \(n^2\), cubes \(n^3\), triangular \(\frac{n(n+1)}{2}\), primes
Back to the quiz
When you are ready, return to the quiz at the top of the page and keep practicing sequences and patterns.
⭐⭐
🔢
Sequences & Patterns
Step-by-step guide
Tap to open ->
Loading...
Sequences & Patterns Lesson
1 / 8
Lesson Overview
Lesson overview
Purpose: Build strong pattern recognition skills and learn the most common number sequence rules used in tests: arithmetic, geometric, recursive, and classic sequences (squares, cubes, triangular numbers, Fibonacci, primes).
Success criteria
Use sequence notation correctly: \(a_1, a_2, a_3,\dots\) and \(a_n\).
Find the next term by checking differences, ratios, and simple operations.
Recognize an arithmetic sequence (constant difference \(d\)) and use \(a_n=a_1+(n-1)d\).
Recognize a geometric sequence (constant ratio \(r\)) and use \(a_n=a_1\cdot r^{\,n-1}\).
Identify common patterns: squares \(n^2\), cubes \(n^3\), triangular numbers \(\frac{n(n+1)}{2}\), Fibonacci-style sequences, and primes.
Write a recursive rule when a pattern is “build from the previous term”.
Check that your rule matches every given term (not just the first two).
Key vocabulary
Sequence: an ordered list of numbers.
Term: one number in the sequence.
Index: the position number (1st term, 2nd term, …, \(n\)th term).
Explicit rule: a formula that gives \(a_n\) directly.
Recursive rule: a rule that defines \(a_n\) from earlier terms (like \(a_n=a_{n-1}+d\)).
Common difference \(d\): constant change between terms in an arithmetic sequence.
Common ratio \(r\): constant multiplier between terms in a geometric sequence.
Quick pre-check
Pre-check 1: In the notation \(a_1, a_2, a_3,\dots\), what does \(a_5\) mean?
Hint: The subscript shows the position in the sequence.
Pre-check 2: The sequence \(4, 7, 10, 13, \dots\) is arithmetic. What is the common difference?
Learning goal: Decide which strategy to use: differences, ratios, or a well-known pattern.
Key idea
When you see a sequence, start with two quick checks:
Differences: compute \(a_{n}-a_{n-1}\). A constant difference suggests an arithmetic sequence.
Ratios: compute \(\dfrac{a_n}{a_{n-1}}\) (when terms are nonzero). A constant ratio suggests a geometric sequence.
If differences are not constant, you can sometimes look at second differences (differences of the differences). For example, square numbers have increasing differences \(3,5,7,\dots\).
Worked example
Example: Consider the sequence \(1, 4, 9, 16, \dots\).
Differences: \(4-1=3\), \(9-4=5\), \(16-9=7\). The differences are the odd numbers \(3,5,7,\dots\), so the pattern is square numbers: \[ a_n = n^2. \](Indeed: \(1^2=1\), \(2^2=4\), \(3^2=9\), \(4^2=16\).)
Try it
Try it 1: What number comes next in the sequence \(3, 6, 12, 24, \dots\)?
Hint: The ratio is constant: each term is multiplied by \(2\).
Try it 2: What number comes next in the sequence \(2, 4, 7, 11, 16, \dots\)?
Hint: Differences are \(+2,+3,+4,+5\). The next difference is \(+6\).
Summary
Check differences for arithmetic patterns.
Check ratios for geometric patterns.
Recognize classic patterns like squares \(n^2\).
Arithmetic Sequences
Arithmetic sequences: constant difference
Learning goal: Find the common difference \(d\) and use the \(n\)th term formula \(a_n=a_1+(n-1)d\).
Key idea
A sequence is arithmetic if you add the same number \(d\) each time: \[ a_n = a_1 + (n-1)d. \] The common difference is \(d=a_2-a_1\) (and should match \(a_3-a_2\), \(a_4-a_3\), etc.).
Worked example
Example: Find the 10th term of \(7, 14, 21, 28, \dots\).
This is arithmetic with \(a_1=7\) and \(d=7\). \[ a_{10}=a_1+(10-1)d = 7 + 9\cdot 7 = 7 + 63 = 70. \]
Try it
Try it 1: What is the 7th term in the sequence defined by \(a_n = 3n - 1\)?
Hint: Substitute \(n=7\): \(a_7=3\cdot 7 - 1\).
Try it 2: Next: \(10, 7, 4, 1, \dots\)
Hint: The difference is constant: \(7-10=-3\), \(4-7=-3\), \(1-4=-3\).
Learning goal: Find the common ratio \(r\) and use the \(n\)th term formula \(a_n=a_1\cdot r^{\,n-1}\).
Key idea
A sequence is geometric if you multiply by the same number \(r\) each time: \[ a_n = a_1 \cdot r^{\,n-1}. \] The common ratio is \(r=\dfrac{a_2}{a_1}\) (and should match \(\dfrac{a_3}{a_2}\), \(\dfrac{a_4}{a_3}\), etc., when terms are nonzero).
Worked example
Example: Find the 5th term of \(2, 6, 18, 54, \dots\).
This is geometric with \(a_1=2\) and \(r=3\). \[ a_5 = a_1 \cdot r^{4} = 2\cdot 3^4 = 2\cdot 81 = 162. \]
Try it
Try it 1: Find the 6th term: \(1, 2, 4, 8, 16, \dots\)
Hint: Each term is multiplied by \(2\).
Try it 2: What is the common ratio in the sequence \(5, 10, 20, 40, \dots\)?
Hint: Compute \(10/5\), \(20/10\), \(40/20\).
Summary
Geometric sequence \(\Rightarrow\) constant ratio \(r\).
\(n\)th term formula: \(a_n=a_1\cdot r^{\,n-1}\).
Classic Sequences
Squares, cubes, Fibonacci, and primes
Learning goal: Recognize classic sequences and know their key rules.
Key idea
Squares: \(1,4,9,16,\dots\) are \(n^2\).
Cubes: \(1,8,27,64,\dots\) are \(n^3\).
Fibonacci-style: each term is the sum of the previous two.
Primes: numbers greater than 1 with exactly two positive divisors (1 and itself).
Worked example
Example: What number comes next in \(1, 8, 27, 64, \dots\)?
These are cubes: \(1^3=1\), \(2^3=8\), \(3^3=27\), \(4^3=64\). So the next term is \(5^3=125\).
Try it
Try it 1: What number comes next in the sequence \(1, 1, 2, 3, 5, \dots\)?
Hint: Add the previous two terms: \(3+5=8\).
Try it 2: What number comes next in the sequence \(2, 3, 5, 7, 11, 13, \dots\)?
Hint: Continue the prime numbers in order.
Summary
Squares: \(a_n=n^2\). Cubes: \(a_n=n^3\).
Fibonacci-style: add previous two terms.
Primes: \(2,3,5,7,11,13,17,\dots\).
Recursive Sequences
Recursive rules and difference patterns
Learning goal: Write a recursive rule and use differences to extend a sequence correctly.
Key idea
A recursive sequence tells you how to get the next term from earlier terms. To define a recursive sequence, you usually need:
a starting value (like \(a_1=1\)), and
a rule (like \(a_n=a_{n-1}+3\) for \(n\ge 2\)).
Worked example
Example: The sequence is \(1, 2, 4, 7, 11, \dots\). Describe a recursive rule and find a formula.
Differences: \(+1, +2, +3, +4, \dots\). A recursive rule is: \[ a_1=1,\quad a_n=a_{n-1}+(n-1)\ \text{for } n\ge 2. \] Because you add \(1+2+\cdots+(n-1)\), an explicit formula is: \[ a_n = 1 + \frac{n(n-1)}{2}. \]
Try it
Try it 1: What number comes next in the sequence \(3, 8, 15, 24, \dots\)?
Hint: Differences are \(+5,+7,+9\). Next difference is \(+11\).
Try it 2: Which recursive rule matches \(1, 2, 4, 8, 16, \dots\)?
Hint: Each term is double the previous term.
Summary
Recursive rules need a start value and a rule for building the next term.
Learning goal: Handle mixed patterns (like “times 2 plus 1”) and confirm your rule fits every term.
Key idea
Some sequences are not purely arithmetic or geometric. A common exam pattern is: multiply then add or subtract. Always verify the rule works from one term to the next.
Note: In real math, many different rules can fit a few starting terms. In school problems, the intended rule is usually the simplest pattern that matches all given terms.
Worked example
Example: What number comes next in \(1, 3, 7, 15, 31, \dots\)?
Each term is “double the previous term, then add 1”: \(1\to 3\) (\(\times 2 + 1\)), \(3\to 7\) (\(\times 2 + 1\)), \(7\to 15\) (\(\times 2 + 1\)). So the next term is \(31\times 2 + 1 = 63\). This also matches the explicit form \(a_n=2^n-1\).
Try it
Try it 1: What number comes next in \(1, 3, 7, 15, 31, \dots\)?
Hint: The pattern is \(\times 2 + 1\).
Try it 2: What is the 7th term in the sequence \(1, 2, 3, 5, 8, 13, \dots\)?
Hint: Each term is the sum of the previous two: \(8+13=21\).
Summary
Mixed patterns often use “multiply then add/subtract”.
Always verify your rule matches each step in the sequence.
Applications & History
Why sequences and patterns matter
Learning goal: Connect sequences to real contexts (growth, schedules, coding) and learn a little history behind famous patterns.
Where you use sequences
Money and planning: arithmetic increases (saving a little more each week) and geometric growth (compound interest).
Science and growth: doubling or halving over time (population, bacteria, decay).
Computing and coding: loops, step sizes, and repeated multiplication.
Puzzles and games: pattern recognition and logic.
Worked example: a growing pattern
Example: A pattern has \(1\) dot in stage 1, \(3\) dots in stage 2, \(6\) dots in stage 3, and \(10\) dots in stage 4. What is stage 5?
Differences are \(+2, +3, +4\), so next is \(+5\). Stage 5 is \(10+5=15\). This is the triangular number pattern: \[ T_n=\frac{n(n+1)}{2}. \] So \(T_5=\frac{5\cdot 6}{2}=15\).
Try it
Try it 1: What is the 5th triangular number \(T_5\)?
Hint: Use \(T_n=\frac{n(n+1)}{2}\) with \(n=5\).
Fun facts (a little history)
Fibonacci: The Fibonacci sequence became famous in Europe through Leonardo of Pisa (Fibonacci). It appears in many models of growth and patterns in nature.
Gauss and sums: A classic story says young Gauss found a fast way to add \(1+2+\cdots+100\). Triangular numbers use the same idea.
Big idea: Sequences are the building blocks for later topics like series, functions, and mathematical modeling.
Try it 2: Which description best matches a geometric sequence?
Hint: Geometric sequences have a constant ratio \(r\).
Recursive rules build the next term from earlier terms.
Next step: Close this lesson and try your quiz again. If you miss a question, reopen the book and review the page that matches the pattern type you need (differences, ratios, arithmetic, geometric, or classic sequences).
Streak 5+
Streak 10+
Streak 15+
Streak 20+
Streak 25+