Arithmetic Sequences

Find common difference and terms in arithmetic sequences. Math 20-1 Alberta curriculum.

Lesson 3.1 of Sequences and Series in Math 20-1 — Alberta curriculum lessons.

What you'll learn in this lesson

• What is a Sequence?. Learn what sequences are, the difference between finite and infinite sequences, and key vocabulary like terms, general term notation, and subscripts. Explore pattern recognition through examples including triangular dot arrangements.
• Arithmetic Sequences. Discover what makes a sequence arithmetic: a constant common difference d between consecutive terms. Analyze sequences to find the first term, common difference, and specific terms. Learn why sequence graphs are discrete dot plots.
• The General Term Formula. Build the powerful formula t_n = t_1 + d(n - 1) from first principles. Use it to find any term directly without listing all previous terms, and flip it around to find the position of any given value.
• Working with the Formula. Master every question type: given two non-consecutive terms, writing a simplified general term, counting terms in a finite sequence, finding unknowns, inserting arithmetic means, and solving real-world applications.

Inside the lesson — a free preview

Definition

What is a Sequence?. A sequence is a list of numbers in a specific order. Each number in the list is called a term.

Sequences come in two types:

Finite — they stop at some point (like 2, 4, 6, 8, 10)

Infinite — they continue forever (like 2, 4, 6, 8, 10, )

The (called an ellipsis) is how we show a sequence keeps going.

Spotting the Pattern

Every sequence follows some rule. The rule might be simple (add 2 each time) or more complex (add the two previous terms together). Look at the sequences below and see if you can find each one's pattern before checking the answer.

What Kinds of Patterns Exist?. Some patterns add or subtract a constant. Others multiply by a constant. Others, like Fibonacci, use a more complex rule.

In this lesson, we focus on the first type: sequences where you add (or subtract) the same number each time.

Worked example: Visual Pattern Example: Triangular Dots

Sometimes patterns appear as pictures. Here's a classic one — triangular dot arrangements.

Finding the Pattern in Dot Counts: 3, 6, 10, 15, . Look at the differences between consecutive terms:

6 - 3 = 3

10 - 6 = 4

15 - 10 = 5

The differences themselves form a pattern: 3, 4, 5, — each new diagram adds one more dot than the previous addition.

To find the 10th term, we keep adding:

5th term: 15 + 6 = 21

6th term: 21 + 7 = 28

7th term: 28 + 8 = 36

8th term: 36 + 9 = 45

9th term: 45 + 10 = 55

10th term: 55 + 11 = 66

The 10th diagram has 66 dots.

Important: This is NOT an Arithmetic Sequence. This pattern is not an arithmetic sequence because the differences keep changing (3, 4, 5, ). True arithmetic sequences have a constant difference. We'll define those formally in the next tab.

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