Introduction to Sequences (Edexcel IGCSE Maths)

Spec point — 3L

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 1^st term **
    • n = 2** refers to the 2^ndterm
    • 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 1^stterm
    • a₂ represents the 2^ndterm
    • 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, ...

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 100^th 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

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 12^th 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 The nth term of a sequence is given by 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 sum of the 7^th term and the 20^th term of the sequence.

Answer:

For the 7^th* term, substitute n = 7 into the formula*

3×7 + 2 = 23

For the 20^th* term, substitute *n = 20 into the formula

3×20 + 2 = 62

Add the two terms together

23 + 62 = 85

The sum of the 7^th term and the 20**^th** term is 85** *

(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 31^st term)

Final answer: 96 is not in the sequence