Sequence (Definition & Examples)

What is a Sequence?

A sequence is an arrangement of numbers that follows a particular pattern or rule e.g. 3, 7, 9, 11, 15… The rule here is that you add 4 to each term.

Example 1.1.1:

Find the next three terms of these sequences. Write down the rule for each sequence.

(i) 4, 9, 14, 19, 24
(ii) 0.25, 0.28, 0.31, 0.34, 0.37…
(iii) 4, -1, -6, -11, -16…

Solution

(i) 4, 9, 14, 19, 24, 29, 34, 39  | rule = add 5

(ii) 0.25, 0.28, 0.31, 0.34, 0.37, 0.40, 0.43, 0.46 | rule = add 0.03

(iii) 4, -1, -6, -11, -16, -21, -26, -31 | rule = add -5

Example 1.1.2:

Find a formula for the nth term of the sequences in example 1 above and use your formula to find the 20^th term for each.

Solution

(i) d = 5, a = 4

since the rule is by adding 5

nth term is given by:

⇒ 5_n = 5, 10, 15, 20, 25

the sequence = 4, 9, 14, 19, 24

the difference = 1  1  1  1  1

The formula is given as 5n – 1

20^th term = 5 × 20 – 1 = 100 – 1

= 99

(ii) rule is add 0.03

⇒ 0.03n = 0.03  0.06  0.09  0.12  0.15

the sequence = 0.25  0.28  0.31  0.34  0.37

difference = 0.22  0.22  0.22  0.22  0.22

Formula = 0.03n + 0.22

or = \( \frac{3n + 22}{100}\)

20^th term = 0.03 × 20 + 0.22

= 0.6 + 0.22

= 0.82

(iii) the rule is add -5

⇒ -5n = -5  -10  -15  -20  -25

the sequence = 4  -1  -6  -11  -16

the difference = 9  9  9  9  9

Formula = -5n + 9

20^th term = -5 × 20 + 9

= -100 + 9

= -91

Example 1.1.3:

Find the 6^th and 24^th terms of the following

(i) 6n – 3n²  
(ii) \( \frac{5n \;-\; 3}{4}\)

Solution

When n = 6

(i) 6n – 3n² = 6 × 6 – 3(6²)

= 36 – 3(36)

i.e. 6n² – 3n² = 36 – 108

= -72

When n = 24

6n – 3n² = 6 × 24 – 3(24²)

= 144 – 3(576)

= 144 – 1728

6n – 3n²= -1584 

(ii) \( \frac{5n \;-\; 3}{4}\)

when n = 6

\( \frac{5n \;-\; 3}{4}\) = \( \frac{5 \; \times \; 6 \;-\; 3}{4}\)

= \( \frac{30 \;-\; 3}{4}\)

= \( \frac{27}{4}\)

= \( \scriptsize 6 \normalsize \frac{3}{4} \scriptsize \; or \; 6.75\)

When n = 24

\( \frac{5n \;-\; 3}{4} \\ = \frac{5 \; \times \; 24 \;-\; 3}{4}\)

= \( \frac{120 \;-\; 3}{4}\)

= \( \frac{117}{4}\)

= \( \scriptsize 29 \normalsize \frac{1}{4}\scriptsize \; or \; 29.25\)