What are Sequences?

A sequence is a list of numbers that follow a particular pattern. Each number in the sequence is called a term. You can think of a sequence like a set of stepping stones leading you to a destination. Each step shows you where to go next!

Types of Sequences

There are many types of sequences, but we will focus on two main types:

  1. Arithmetic Sequences
  2. Geometric Sequences

Arithmetic Sequences

What is an Arithmetic Sequence?

An arithmetic sequence is a sequence of numbers where each term is found by adding the same amount (called the “common difference”) to the previous term.

Example:

  • 2, 4, 6, 8, 10

Here, the common difference is 2 because we add 2 to each term to get the next one.

Key Rules:

  • Identify the first term.
  • Find the common difference by subtracting one term from the next.
  • Continue adding the common difference to find more terms.

Tips and Tricks:

  • Use a number line to help you visualize the jumps.
  • Write down the first few terms to see the pattern more clearly.

Geometric Sequences

What is a Geometric Sequence?

A geometric sequence is a sequence of numbers where each term is found by multiplying the previous term by the same number (called the “common ratio”).

Example:

  • 3, 6, 12, 24, 48

Here, the common ratio is 2 because we multiply each term by 2 to get the next one.

Key Rules:

  • Identify the first term.
  • Find the common ratio by dividing one term by the previous term.
  • Continue multiplying by the common ratio to find more terms.

Tips and Tricks:

  • Draw a tree diagram to see how each term branches out.
  • Keep track of your multiplication to avoid mistakes.

Questions

Easy Level (1-20)

  1. What is the first term in the sequence: 5, 10, 15, 20?
  2. What is the common difference in the sequence: 1, 3, 5, 7?
  3. Write the next term in the sequence: 10, 20, 30, ___?
  4. What is the first term of the sequence: 2, 4, 6, 8?
  5. What is the 5th term in the arithmetic sequence: 4, 8, 12, 16, ___?
  6. Identify the next term: 3, 6, 9, ___?
  7. What is the common difference in the sequence: 7, 10, 13?
  8. Write the next term in the sequence: 1, 2, 4, 8, ___?
  9. What is the first term of the sequence: 0, 5, 10, 15?
  10. Find the 4th term in the sequence: 5, 10, 15, ___?

Medium Level (21-40)

  1. What is the common difference in the sequence: 12, 9, 6, 3?
  2. Write the next two terms in the sequence: 2, 4, 8, 16, ___, ___?
  3. What is the 6th term in the arithmetic sequence: 3, 6, 9, ___?
  4. Identify the common ratio in the sequence: 1, 2, 4, 8?
  5. Write the first four terms of the sequence starting from 7 with a common difference of 3.
  6. What is the 3rd term in the sequence: 5, 10, 20, ___?
  7. Identify the next term: 1, 2, 4, 8, 16, ___?
  8. What is the common difference in the sequence: 10, 20, 30, 40?
  9. Write the next term in the geometric sequence: 4, 12, 36, ___?
  10. Find the common ratio in the sequence: 2, 6, 18, ___?

Hard Level (41-60)

  1. What is the 10th term in the arithmetic sequence: 2, 5, 8, ___?
  2. Write the first five terms of a sequence that starts at 10 and has a common ratio of 2.
  3. In the sequence 5, 10, 15, 20, what would be the 12th term?
  4. If the common difference is 4, what is the 7th term in the sequence starting with 3?
  5. Write the next three terms in the sequence: 1, 4, 9, 16, ___, ___, ___?
  6. What is the common ratio in the sequence: 3, 9, 27, ___?
  7. If the 5th term in an arithmetic sequence is 25 and the common difference is 5, what is the first term?
  8. Identify the next term in the sequence: 100, 50, 25, ___?
  9. What is the common difference in the sequence 30, 25, 20, 15?
  10. Write the first five terms of a geometric sequence that starts with 1 and has a common ratio of 3.

Answers

Easy Level Answers:

  1. 5
  2. 2
  3. 40
  4. 2
  5. 20
  6. 12
  7. -3
  8. 16, 32
  9. 0
  10. 20

Medium Level Answers:

  1. -3
  2. 32, 64
  3. 18
  4. 2
  5. 7, 10, 13, 16
  6. 40
  7. 32
  8. 10
  9. 108
  10. 54

Hard Level Answers:

  1. 29
  2. 10, 20, 40, 80, 160
  3. 50
  4. 27
  5. 25, 36, 49
  6. 3
  7. 10
  8. 12.5
  9. -5
  10. 1, 3, 9, 27, 81

With this guide, you should be well on your way to understanding sequences! Remember, practice makes perfect!