math 10

• This module will teach you how to deal with patterns, specifically, number patterns. These number patterns are called sequences.
• As you go over the lessons and activities, you will be able to learn how to generate patterns, illustrate arithmetic sequence and to find its nth term.
• Most Essential Learning Competencies

  • Generates patterns
  • Illustrates arithmetic sequence
  • Determines arithmetic means, nth term of an arithmetic sequence and sum of the terms of a given arithmetic sequence
• Do not write anything on this module. Use your notebook and a sheet of paper for your answers to every activity. All answer sheets will be collected by your teacher.
• What I Know


  1. Read each item carefully

  2. Choose the letter of the correct answer

  3. Write it on a separate sheet of paper
• Arithmetic sequence


  A sequence where the difference between any two consecutive terms is constant
• This module will help you understand more important concepts on patterns that you can apply in your daily life.
• What's New


  Learn how to generate patterns in numbers
• Patterns can be seen in shapes and in nature. Patterns generated in the pictures and the number of seats in the auditorium are examples of sequences.
• What is It


  Complete the pattern generated from the activity
• Finite sequence


  A sequence with a last term
• Infinite sequence


  A sequence that continues indefinitely
• Activity 2: Finite or Infinite?


  Identify whether the sequence is finite or infinite
• A sequence is said to be finite if its domain is the set {1, 2, 3,...,n} and is said to be infinite if its domain is the set of positive integers.
• General term


  The function that generates the terms of a sequence
• Activity 3: Term after Term

  1. Find the first five terms of sequences with given general terms

  2. Find the 15th term of a sequence with a given general term

  3. Find the 10th term of a given arithmetic sequence
• 2n + 1


  General term of the sequence
• Finding the first 5 terms of the sequence

  1. For n=1, a1=2(1)+1=3

  2. For n=2, a2=2(2)+1=5

  3. For n=3, a3=2(3)+1=7

  4. For n=4, a4=2(4)+1=9

  5. For n=5, a5=2(5)+1=11
• Suppose n=12, then, a12=2(12)+1=25
• Generating number sequences is easy
• Find the first five terms of the sequence whose general term is

  • an =3n
  • an =10-2n
• The first five terms of the sequence an =3n are 3,9,27,81,243
• The first five terms of the sequence an =10-2n are 8,6,4,2,0
• The 15th term of the sequence an = n+1/n is 16/15
• The 10th term of the sequence 1,5,9,13,... is 37
• Generating a rule for a sequence from the first few terms is not always easy
• Find a formula for the nth term of the sequence 2, 11, 26, 47, ...


  an = 3n^2 - 1
• Find the general term for the sequence 4, 5/8, 6/27, 7/64,...

  an = (n+3)/n^3
• Find the general term of the sequence whose first five terms are 1/2, 3/4, 5/6, 7/8, 9/10,...

  an = (2n-1)/2n
• Find the nth term of the sequence -2, 4, -8, 16, -32, 64...

  an = -2^n
• Finite sequences have a domain of {1, 2, 3, ..., n} where n is a natural number
• Infinite sequences have a domain of the set of positive integers
• Generating patterns is important in problem solving
• Advantages of generating a rule in a sequence include being able to find any term in the sequence and understand the underlying structure
• Situations where generating a rule can help solve problems

  • Predicting future values in a sequence
  • Optimizing a process with a pattern
  • Identifying trends in data
• Sequence numbers


  • 36
  • 9, 89, 86, 83, 80
  • 1.2, 1.8, 2.4
  • 2/3, 1, 4/3, 5/3, 2
• Activity 2: Arithmetic or Not?

  1. Identify whether the sequence is arithmetic or not

  2. If it is, find the common difference and the next three terms
• To find the nth term of an arithmetic sequence, a formula would help
• Arithmetic sequence


  A sequence where every term after the first is obtained by adding a constant called the common difference
• Finding the nth term of an arithmetic sequence

  Use the formula: an = a1 + (n-1)d, where an is the nth term, a1 is the 1st term, and d is the common difference