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Intro to arithmetic sequences

Get comfortable with sequences in general, and learn what arithmetic sequences are.
Before you take this lesson, make sure you know how to add and subtract negative numbers.

What is a sequence?

Here are a few lists of numbers:
  • 3, 5, 7 ...
  • 21, 16, 11, 6 ...
  • 1, 2, 4, 8 ...
Ordered lists of numbers like these are called sequences. Each number in a sequence is called a term.
3, comma5, comma7, comma, point, point, point
\uparrow\uparrow\uparrow
1, start superscript, start text, s, t, end text, end superscript, start text, space, t, e, r, m, end text2, start superscript, start text, n, d, end text, end superscript, start text, space, t, e, r, m, end text3, start superscript, start text, r, d, end text, end superscript, start text, space, t, e, r, m, end text
Sequences usually have patterns that allow us to predict what the next term might be.
For example, in the sequence 3, 5, 7 ..., you always add two to get the next term:
start color #ed5fa6, plus, 2, \curvearrowright, end color #ed5fa6start color #ed5fa6, plus, 2, \curvearrowright, end color #ed5fa6
3, comma5, comma7, comma, point, point, point
The three dots that come at the end indicate that the sequence can be extended, even though we only see a few terms.
We can do so by using the pattern.
For example, the fourth term of the sequence should be nine, the fifth term should be 11, etc.
start color #ed5fa6, plus, 2, \curvearrowright, end color #ed5fa6start color #ed5fa6, plus, 2, \curvearrowright, end color #ed5fa6start color #ed5fa6, plus, 2, \curvearrowright, end color #ed5fa6start color #ed5fa6, plus, 2, \curvearrowright, end color #ed5fa6
3, comma5, comma7, comma9, comma11, comma, point, point, point

Check your understanding

Extend the sequences according to their pattern.
Problem 1
Pattern: Add five to the previous term.
start color #ed5fa6, plus, 5, \curvearrowright, end color #ed5fa6start color #ed5fa6, plus, 5, \curvearrowright, end color #ed5fa6start color #ed5fa6, plus, 5, \curvearrowright, end color #ed5fa6
3, comma8, comma13, comma
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
comma, point, point, point

Problem 2
Pattern: Subtract three from the previous term.
start color #ed5fa6, minus, 3, \curvearrowright, end color #ed5fa6start color #ed5fa6, minus, 3, \curvearrowright, end color #ed5fa6start color #ed5fa6, minus, 3, \curvearrowright, end color #ed5fa6
20, comma17, comma14, comma
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
comma, point, point, point

Problem 3
Pattern: Multiply the previous term by two.
start color #ed5fa6, times, 2, \curvearrowright, end color #ed5fa6start color #ed5fa6, times, 2, \curvearrowright, end color #ed5fa6start color #ed5fa6, times, 2, \curvearrowright, end color #ed5fa6
3, comma6, comma12, comma
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
comma, point, point, point

Problem 4
Match each sequence with its pattern.

What is an arithmetic sequence?

For many of the examples above, the pattern involves adding or subtracting a number to each term to get the next term. Sequences with such patterns are called arithmetic sequences.
In an arithmetic sequence, the difference between consecutive terms is always the same.
For example, the sequence 3, 5, 7, 9 ... is arithmetic because the difference between consecutive terms is always two.
start color #ed5fa6, plus, 2, \curvearrowright, end color #ed5fa6start color #ed5fa6, plus, 2, \curvearrowright, end color #ed5fa6start color #ed5fa6, plus, 2, \curvearrowright, end color #ed5fa6
3, comma5, comma7, comma9, comma, point, point, point
The sequence 21, 16, 11, 6 ... is arithmetic as well because the difference between consecutive terms is always minus five.
start color #ed5fa6, minus, 5, \curvearrowright, end color #ed5fa6start color #ed5fa6, minus, 5, \curvearrowright, end color #ed5fa6start color #ed5fa6, minus, 5, \curvearrowright, end color #ed5fa6
21, comma16, comma11, comma6, comma, point, point, point
The sequence 1, 2, 4, 8 ... is not arithmetic because the difference between consecutive terms is not the same.
start color #ed5fa6, plus, 1, \curvearrowright, end color #ed5fa6start color #ed5fa6, plus, 2, \curvearrowright, end color #ed5fa6start color #ed5fa6, plus, 4, \curvearrowright, end color #ed5fa6
1, comma2, comma4, comma8, comma, point, point, point

Check your understanding

Problem 5
Select all arithmetic sequences.
Choose all answers that apply:
Choose all answers that apply:

Problem 6
The first term of a sequence is one. Which of the following patterns would make the sequence arithmetic?
Choose all answers that apply:
Choose all answers that apply:

The common difference

The common difference of an arithmetic sequence is the constant difference between consecutive terms.
For example, the common difference of 10, 21, 32, 43 ... is 11:
start color #ed5fa6, plus, 11, \curvearrowright, end color #ed5fa6start color #ed5fa6, plus, 11, \curvearrowright, end color #ed5fa6start color #ed5fa6, plus, 11, \curvearrowright, end color #ed5fa6
10, comma21, comma32, comma43, comma, point, point, point
The common difference of –2, –5, –8, –11 ... is negative three:
start color #ed5fa6, minus, 3, \curvearrowright, end color #ed5fa6start color #ed5fa6, minus, 3, \curvearrowright, end color #ed5fa6start color #ed5fa6, minus, 3, \curvearrowright, end color #ed5fa6
minus, 2, commaminus, 5, commaminus, 8, commaminus, 11, comma, point, point, point

Check your understanding

Problem 7
What is the common difference of 2, 8, 14, 20 ...?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

Problem 8
What is the common difference of 5, comma, 2, comma, minus, 1, comma, minus, 4, point, point, point?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

Problem 9
What is the common difference of 1, comma, 1, start fraction, 1, divided by, 3, end fraction, comma, 1, start fraction, 2, divided by, 3, end fraction, comma, 2, comma, point, point, point?
  • Your answer should be
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4

Reflection question
What must be true about an arithmetic sequence whose common difference is negative?
Choose 1 answer:
Choose 1 answer:

Challenge problem
The first term of an arithmetic sequence is 10 and its common difference is negative seven.
What is the fourth term of the sequence?
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text

What's next?

Learn about formulas of arithmetic sequences, which give us the information we need to find any term in the sequence.

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