Sequences

A set of numbers where each number can be obtained from the previous number according to some rule is called a sequence The most simplest sequence is the set of natural numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, . . . The terms of other sequences can be obtained from the sequence of natural numbers

Consider the even numbers 1st 2nd 3rd 4th 5th 6th, … nth 1, 2, 3, 4, 5, 6, … 2, 4, 6, 8, 10, 12, … Can you describe how the even numbers are obtained from the sequence of natural numbers? We can see that the even numbers can be obtained by multiplying the natural numbers by 2 The nth term is obtained by 2n

Consider the odd numbers 1st 2nd 3rd 4th 5th 6th, … nth 1, 2, 3, 4, 5, 6, … 1, 3, 5, 7, 9, 11, … Can you describe how the odd numbers are obtained from the sequence of natural numbers? We can see that the odd numbers can be obtained by multiplying the natural numbers by 2 and then subtracting 1 The nth term is obtained by 2n - 1

Write down the first 5 terms of the sequence produced by multiplying the natural numbers by 7 and subtracting 4 1, 2, 3, 4, 5, … 7 - 4, 14 - 4, 21 - 4, 28 - 4 , 35 - 4, … 3, 10, 17, 24, 31, … What is the nth term of this sequence? The nth term is given by 7n - 4

Describe the way that this sequence is produced from the natural numbers: 5, 8, 11, 14, … First number the terms of the sequence and find the difference between each term 1st 2nd 3rd 4th 5, 8, 11, 14, … + 3 + 3 + 3 Because the difference is 3, this sequence will be based on the 3 times table

1st 2nd 3rd 4th nth 3 6 9 12 3n 5, 8, 11, 14, … 3n + 2 The sequence is produced by multiplying the natural numbers by 3 and adding 2 What is the nth term of this sequence? The nth term is given by 3n + 2

Find a rule for the nth term of this sequence: -5, -1, 3 , 7, 11, … First number the terms of the sequence and find the difference between each term 1st 2nd 3rd 4th 5th -5, -1, 3, 7, 11, … + 4 + 4 + 4 + 4 Because the difference is 4, this sequence will be based on the 4 times table

1st 2nd 3rd 4th 5th nth 4 8 12 16 20 4n -5, -1, 3, 7, 11, … 4n - 9 The sequence is produced by multiplying the natural numbers by 4 and subtracting 9 What is the nth term of this sequence? The nth term is given by 4n - 9

A bead maker has red beads and blues beads and will make necklaces with the following pattern: Find a rule for how many blue beads will be used with n red beads. How many blue beads will be needed when there are 100 red beads?

The number of red beads is in the sequence of natural numbers Blue beads 6 8 10 + 2 + 2 The sequence is based on the 2 times table Red beads 1 2 3 2 4 6 Blue beads 6 8 10

Red beads: 1 2 3 n 2 4 6 2n Blue beads: 6 8 10 2n + 4 Number of blue beads is 2 X number of red beads + 4 Number of blue beads = 2n + 4 Number of blue beads = 2 X 100 + 4 = 204