Introduction to Sequences (Edexcel IGCSE Maths)

Spec point — 3.1A

Introduction to sequences

What are sequences?

• A sequence is an ordered set of numbers that follow a rule

  • For example 3, 6, 9, 12...

    • The rule is to add 3 each time

• Each number in a sequence is called a term

• The location of a term within a sequence is called its position

  • The letter n is used for position

    • n = 1 **refers to the 1st term
    • n = 2** refers to the 2nd term
    • If you do not know its position, you can say the n th term

• Another way to show the position of a term is using subscripts

  • A general sequence is given by a₁, a₂, a₃, ...

    • a₁ represents the 1st term
    • a₂ represents the 2nd term
    • a_n represents the nth term

How do I write out a sequence using a term-to-term rule?

• Term-to-term rules tell you how to get the next term from the term you are on

  • It is what you do each time

  • For example, starting on 4, add 10 each time

    • 4, 14, 24, 34, ...

• The common term-to-term rules are:

  • adding or subtracting a number

    • 3, 7, 11, 15, ... the rule is add 4 each time

  • multiplying or dividing by a number

    • 2, 6, 18, 54, ... the rule is multiply by 3 each time

How do I write out a sequence using a position-to-term rule?

• A position-to-term rule is an algebraic expression in n that lets you find any term in the sequence

  • This is also called the n th term formula

• You need to know what position in the sequence you are looking for

  • To get the 1st term, substitute in n = 1 **
  • To get the 2nd term, substitute in n = 2**

• You can jump straight to the 100th term by substituting in n = 100

  • You do not need to find all 99 previous terms

• For example, the n th term is 8n + 2

  • The 1st term is 8×1 + 2 = 10
  • The 2nd term is 8×2 + 2 = 18
  • The 100th term is 8×100 + 2 = 802

• For example, the n th term is n² + 1

  • The 1st term is 1 ² + 1 = 2
  • The 2nd term is 2² + 1 = 5
  • The 100th term is 100² + 1 = 10 001

How do I know if a value belongs to a sequence?

• If you know the n th term formula, set the value equal to the formula

  • This creates an equation to solve for n

• For example, a sequence has the n th term formula 8*n * + 2

  • Is 98 in the sequence? $\begin{gathered} 8 n + 2 & = 98 \\ 8 n & = 96 \\ n = \frac{96}{8} \\ n = 12 \end{gathered}$

    • It is in the sequence, it is the 12th term

  • Is 124 in the sequence? $\begin{gathered} 8 n + 2 & = 124 \\ 8 n & = 122 \\ n = \frac{122}{8} \\ n = 15.25 \end{gathered}$

    • n is not a whole number, so it is not in the sequence

Exam Hint In the exam, it helps to write the position number (the value of n) above each term in the sequence.

Worked Example A sequence has the n th term formula 3n + 2.

(a) Find the first three terms in the sequence.

Answer:

Substitute n = 1, n = 2 and n = 3 into the formula

3×1 + 2 = 5

3×2 + 2 = 8

3×3 + 2 = 11

Final answer: 5, 8, 11

(b) Find the 80th term.

Answer:

Substitute n = 80 into the formula

3×80 + 2

The 80th term is 242 *

(c) Determine whether the number 96 is in the sequence.

Answer:

Set the formula equal to 96

$3n+2=96$

Solve to find n If n is a whole number, it is a term in the sequence

$\begin{array}{rcl} 3 n & = 96 - 2 \\ 3 n & = 94 \\ n = \frac{94}{3} \end{array}$

94 is not divisible by 3 The nearest possible value is 95 ((95-2) ÷ 3 = 31, the 31st term)

Final answer: 96 is not in the sequence