Finding Formulae

1
Finding Formulae

• Linear Quadratic

2
Sequences Series

• Sequence
• This is a set of numbers that have a rule to
  connect one number with the next number
• e.g. 1, 3, 5, 7,

• Series
• This is the sum of a number sequence
• e.g. 1 3 5 7

3
Linear Sequences

• Linear Sequence
• This is a sequence where the difference between
  successive terms is constant
• It can either be positive or negative

• For example, in the sequence
• 1, 3, 5, 7,
• The constant difference is 2
• ? First part is 2n
• To get the nth term, subtract 1
• nth term is 2n - 1

4
Linear Sequences

• Example 1
• Find the nth term
• 7, 11, 15, 19,
• ? First Part 4n
• To get the nth term, add 3
• nth term is 4n 3

• Example 2
• Find the nth term
• 32, 29, 26, 23,
• ?? First Part 3n
• To get the nth term, add 35
• nth term is -3n 35
• or 35 3n

5
Linear Sequences

• Find the nth terms of the following sequences
• 1) 11, 18, 25, 32,
• 2) 5, -1, 3, 7,
• 3) 41, 33, 25, 17,
• 4) 17, 8, -1, -10,

6
Linear Sequences - Answers

• 1) 11, 18, 25, 32,
• nth term 7n 4
• 2) 5, -1, 3, 7,
• nth term 4n - 9
• 3) 41, 33, 25, 17,
• nth term 49 - 8n
• 4) 17, 8, -1, -10,
• nth term 26 - 9n

7
Quadratic Sequences

• Quadratic Sequences
• This is a sequence in which the first difference
  is NOT constant
• The second difference has to be taken in order to
  get a constant difference and this gives a term
  in n2
• The coefficient of the n2 term is found by
• second difference
• 2

8
Quadratic Sequences

• Quadratic Sequences
• For example, in the sequence 2, 6, 12, 20, 30,
• The first differences are 4, 6, 8, 10, etc.
• The second constant difference is 2
• ?The nth term is of the form 1n2
• On subtracting the value of n2 from each term,
  the linear sequence 1, 2, 3, 4, 5 is obtained
  which is n
• The sequence is n2 n

9
Quadratic Sequences

• Example 2 Sequence 5, 12, 23, 38, 57,
• The first differences are 7, 11, 15, 19, etc.
• The second constant difference is 4
• ?The nth term is of the form 2n2
• On subtracting the value of 2n2 from each term,
  the linear sequence 3, 4, 5, 6, 7 is obtained
  which is n 2
• The sequence is 2n2 n 2

10
Quadratic Sequences

• Find the nth terms of the following sequences
• 1) 4, 10, 18, 28, 40,
• 2) -1, 0, 3, 8, 15,
• 3) 6, 15, 28, 45, 66,
• 4) 5, 12, 25, 44, 69,

11
Quadratic Sequences

• Find the nth terms of the following sequences
• 1) 4, 10, 18, 28, 40,
• nth term n2 3n
• 2) -1, 0, 3, 8, 15,
• nth term n2 - 2n
• 3) 6, 15, 28, 45, 66,
• nth term 2n2 3n 1
• 4) 5, 12, 25, 44, 69,
• nth term 3n2 - 2n 4

12
Linear Quadratic Sequences

• GCSE Maths For Higher
• Linear Sequences
• Chapter 17
• Exercises 17B 17C
• Quadratic Sequences
• Chapter 17
• Exercise 17D

• Further work
• www.speters.org.uk/maths
• Grade C
• Revision
• Examination Practice
• Grade A
• Revision
• Examination Practice