Learn Sequences & Patterns: Interactive Problems and Worked Examples

How to recognize number patterns quickly

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

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Summary

• Check differences for arithmetic patterns.
• Check ratios for geometric patterns.
• Recognize classic patterns like squares \(n^2\).

Arithmetic sequences: constant difference

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

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Summary

• Arithmetic sequence \(\Rightarrow\) constant difference \(d\).
• \(n\)th term formula: \(a_n=a_1+(n-1)d\).

Geometric sequences: constant ratio

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

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Summary

• Geometric sequence \(\Rightarrow\) constant ratio \(r\).
• \(n\)th term formula: \(a_n=a_1\cdot r^{\,n-1}\).

Squares, cubes, Fibonacci, and primes

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

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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 rules and difference patterns

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 \(n\ge 2\)).

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Summary

• Recursive rules need a start value and a rule for building the next term.
• Differences can reveal “add 1, add 2, add 3, …” patterns.

Mixed patterns and “best rule” thinking

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

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Summary

• Mixed patterns often use “multiply then add/subtract”.
• Always verify your rule matches each step in the sequence.

Why sequences and patterns matter

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

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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.

Final recap

• Arithmetic: constant difference \(d\), \(a_n=a_1+(n-1)d\).
• Geometric: constant ratio \(r\), \(a_n=a_1\cdot r^{\,n-1}\).
• Differences and ratios are the fastest first checks.
• Classic patterns: squares \(n^2\), cubes \(n^3\), triangular \(\frac{n(n+1)}{2}\), Fibonacci, primes.
• Recursive rules build the next term from earlier terms.