Lesson 61: Sequences

New Concept

A sequence is an ordered list of numbers, called terms, that follow a certain pattern or rule, such as arithmetic or geometric progressions.

What’s next

Now that you know the basics, you'll learn to spot different types of sequences and write formulas to describe their powerful patterns.

Property

An arithmetic sequence has a constant difference between terms. To find it, subtract any term from the one that follows it: $6-3=3$, $9-6=3$.

Examples

• In the sequence $5, 10, 15, 20, \dots$, the constant difference is $10-5=5$.
• For the sequence $100, 90, 80, \dots$, the constant difference is $-10$. The next two terms are $70$ and $60$.
• The graph of the sequence $2, 4, 6, \dots$ includes the points $(1, 2), (2, 4), (3, 6)$, which form a straight line.

Explanation

Think of an arithmetic sequence like climbing a staircase where every step is the same height! You just add the same number, the 'constant difference,' over and over to get the next term. Because the change is steady, if you graph the sequence, the points line up perfectly straight, making it super easy to predict what’s next.

Property

A geometric sequence has a constant ratio between terms. To find it, divide any term by the one that precedes it: $\frac{9}{3}=3$, $\frac{27}{9}=3$.

Examples

• In the sequence $2, 6, 18, 54, \dots$, the constant ratio is $\frac{6}{2}=3$.
• For the sequence $100, 50, 25, \dots$, the constant ratio is $\frac{1}{2}$. The next term is $12.5$.
• To find the next term of $4, 16, 64, \dots$, you multiply by the ratio 4: $64 \times 4 = 256$.

Explanation

This sequence is all about multiplication! Each term is found by multiplying the previous one by a fixed number, the 'constant ratio.' This makes the sequence grow incredibly fast, like a viral video. When you graph these sequences, they don't make a straight line but a cool curve that shoots upward, showing that explosive, geometric growth.