Introduction to Sequences

Today, we are going to learn about two important types of sequences: arithmetic sequences and geometric sequences. Understanding these sequences will help you in many areas of maths!

What is a Sequence?

A sequence is simply a list of numbers that follow a specific rule. Each number in the sequence is called a term.

Arithmetic Sequences

What is an Arithmetic Sequence?

An arithmetic sequence is a sequence where each term is found by adding the same number to the previous term. This number is called the common difference.

Key Rules for Arithmetic Sequences

  1. Common Difference: The difference between consecutive terms is constant.
  2. Formula: If the first term is (a_1) and the common difference is (d), the (n)-th term can be found using the formula: a_n = a_1 + (n – 1) \times d

Example

Let’s look at an example of an arithmetic sequence:

  • Sequence: 2, 5, 8, 11, 14, …
  • Common difference ((d)): 3 (because (5 – 2 = 3))

Tips for Arithmetic Sequences

  • Always find the difference between the first two terms.
  • Check if the difference is the same for the next pairs of terms.

Geometric Sequences

What is a Geometric Sequence?

A geometric sequence is a sequence where each term is found by multiplying the previous term by the same number. This number is called the common ratio.

Key Rules for Geometric Sequences

  1. Common Ratio: The ratio between consecutive terms is constant.
  2. Formula: If the first term is (a_1) and the common ratio is (r), the (n)-th term can be found using the formula: a_n = a_1 \times r^{(n – 1)}

Example

Let’s look at a geometric sequence:

  • Sequence: 3, 6, 12, 24, 48, …
  • Common ratio ((r)): 2 (because (6 \div 3 = 2))

Tips for Geometric Sequences

  • Always find the ratio between the first two terms.
  • Check if the ratio is the same for the next pairs of terms.

Summary

To identify whether a sequence is arithmetic or geometric:

  • Arithmetic: Look for a common difference (addition).
  • Geometric: Look for a common ratio (multiplication).

Questions

Easy Level

  1. What is the common difference in the sequence 4, 7, 10?
  2. Is the sequence 1, 3, 5, 7 arithmetic or geometric?
  3. Find the next term in the sequence 10, 15, 20.
  4. What is the 5th term in the arithmetic sequence 2, 5, 8, …?
  5. Identify the common ratio in the sequence 1, 2, 4, 8.
  6. Is the sequence 3, 6, 12, 24 arithmetic or geometric?
  7. What is the 3rd term in the arithmetic sequence 5, 10, 15, …?
  8. Find the next term in the sequence 8, 4, 2.
  9. What is the common difference in the sequence 6, 9, 12?
  10. Identify if the sequence 2, 4, 8, 16 is arithmetic or geometric.

Medium Level

  1. What is the 7th term in the arithmetic sequence 3, 6, 9, …?
  2. Find the common difference in the sequence 11, 8, 5, 2.
  3. What is the 4th term in the geometric sequence 2, 6, 18, …?
  4. Is the sequence 5, 10, 20, 40 arithmetic or geometric? Explain.
  5. Identify the 5th term in the arithmetic sequence 1, 4, 7, …?
  6. Calculate the common ratio for the sequence 5, 15, 45.
  7. What is the next term in the sequence 1, 1, 2, 3, 5?
  8. Is the sequence 7, 14, 21, 28 arithmetic or geometric?
  9. What is the 6th term in the geometric sequence 1, 3, 9, 27, …?
  10. Identify the common difference in the sequence 15, 10, 5.

Hard Level

  1. Find the 10th term of the arithmetic sequence where the first term is 2 and the common difference is 3.
  2. If the 3rd term of a geometric sequence is 12 and the first term is 3, find the common ratio.
  3. Identify the 8th term in the arithmetic sequence 10, 15, 20, …?
  4. Determine if the sequence 4, 8, 16, 32, … is arithmetic or geometric.
  5. If the first term of an arithmetic sequence is 5 and the common difference is 2, find the 12th term.
  6. Examine the sequence 3, 9, 27, 81 and find the common ratio.
  7. What is the 5th term in an arithmetic sequence that starts with 1 and has a common difference of 4?
  8. Identify the 6th term in the geometric sequence 2, 6, 18, …?
  9. If the 7th term of a geometric sequence is 128 and the first term is 2, find the common ratio.
  10. Determine the common difference for the sequence 20, 15, 10, 5.

Answers

Easy Level Answers

  1. 3
  2. Arithmetic
  3. 25
  4. 14
  5. 2
  6. Geometric
  7. 15
  8. 1
  9. 3
  10. Geometric

Medium Level Answers

  1. 21
  2. -3
  3. 54
  4. Geometric; it multiplies by 2.
  5. 13
  6. 3
  7. 8
  8. Arithmetic
  9. 243
  10. -5

Hard Level Answers

  1. 32
  2. 4
  3. 40
  4. Geometric
  5. 25
  6. 3
  7. 17
  8. 54
  9. 4
  10. -5

I hope this helps you understand arithmetic and geometric sequences better! If you have any questions, feel free to ask!