iGCSE (2021 Edition)

# 9.01 Sequences

Lesson

An ordered collection of numbers (or objects) is called a sequence or progression. It is similar to a set, but may contain repetition and the order of the numbers is important. For example $-3,5,13,21...$3,5,13,21... and $1,10,100,1000...$1,10,100,1000... are two interesting mathematical progressions.

If the sequence ends, it is known as a finite sequence. If the sequence continues indefinitely it is said to be infinite.

The elements of a sequence are called terms. For the sequence $2,3,5,7,11,...$2,3,5,7,11,... (the sequence of prime numbers) the first term is $2$2, the second term is $3$3 and term $3$3 is $5$5. We can show this with the following notation $t_1=2$t1=2, $t_2=3$t2=3, $t_3=5$t3=5, and so on. Term $6$6 of this sequence is $13$13, $t_6=13$t6=13.

To refer generally to the $n$nth term we use the notation $t_n$tn. The $n$n is a variable which represents the position of a term in the sequence. If we let $n=1$n=1 then $t_n$tn is $t_1$t1, if we let $n=6$n=6 then $t_n$tn is $t_6$t6. For any term using this notation $t_n$tn, we can then see that the term before would have to be $t_{n-1}$tn1 and the term after $t_n$tn is $t_{n+1}$tn+1.

Sequences do not necessarily have to have a pattern or simple rule for generating terms in the sequence. We could create the sequence $3,1,4,1,5,9,...$3,1,4,1,5,9,... by separating the digits of $\pi$π, we could use a dice to generate a sequence of of rolls $2,1,1,3,4,...$2,1,1,3,4,... or we could write down the prime numbers in order. However, many sequences have a generating rule that can be expressed as a formula, let's look at a couple of different ways of describing such sequences.

### Explicit rules

The recursive rule is limited in that it relies on the previous term in order to find the next term. Consider the sequence $3,5,7,9...$3,5,7,9... We can continue the sequence and find later terms by following this recursive rule however if we wanted to know a much later term, $t_{43}$t43, for example, it would be helpful to have a rule which allows us to find any term and does not rely on knowing the term before to find it.

This rule is called the explicit rule or the general rule and it is written in terms of $n$n. For this sequence the explicit rule would be $t_n=2n+1$tn=2n+1. We can see if we wanted to find the $5$5th term we can substitute $5$5 into the place of $n$n, $t_5=2\times5+1=11$t5=2×5+1=11. By continuing the pattern seen in the sequence $3,5,7,9$3,5,7,9 ... we can see that the $5$5th term is $11$11. We can use this rule to jump to a later term like $t_{43}$t43$t_{43}=2\times43+1=87$t43=2×43+1=87.

#### Worked example

Write the explicit rule to describe the $n$nth term of the sequence $5,9,13,17,21,...$5,9,13,17,21,...

Think: To obtain the explicit rule it can sometimes be useful to look at the sequence as a table of values and try to see the connection between $n$n and the term:

Do

$n$n $t_n$tn Pattern
$1$1 $5$5 $5$5
$2$2 $9$9 $5+4$5+4
$3$3 $13$13 $5+2\times4$5+2×4
$4$4 $17$17 $5+3\times4$5+3×4
...
$n$n $t_n$tn $5+(n-1)\times4=1+4n$5+(n1)×4=1+4n

Hence, the sequence can be written as $t_n=1+4n$tn=1+4n.

#### Practice questions

##### Question 1

If $T_n$Tn describes the $n$nth term in the following sequence, what is $T_3$T3?

$4,-5,6,-7,8,\ldots$4,5,6,7,8,

##### Question 2

Using the following explicit rule, state the first $5$5 terms of the sequence in order starting with $n=1$n=1.

$s_n=n^2+6$sn=n2+6

1. Enter each term on the same line, separated by commas.

## Sequences with technology

A graphics calculator can also be used to generate the terms of a sequence given an explicit rule. Consider the following rule for the $n$nth term of the sequence, $T_n=n^2+1$Tn=n2+1 and see below to view the problem and steps to follow with a calculator.

TI Nspire

How to use the TI Nspire to complete the following tasks regarding explicit sequences

1. Generate the first $10$10 terms of the sequence with the explicit rule: $t_n=n^2+1$tn=n2+1.

2. Find the $100$100th term of the given sequence.

3. Find the sum of the first $10$10 terms.

4. Plot the sequence and describe the long-term behaviour.

#### Practice questions

##### Question 3

For the following explicit rule which starts at $n=1$n=1, what is the sum of the first $50$50 terms?

$T_n=0.7n-5$Tn=0.7n5

##### Question 4

Consider the following sequence starting at $n=1$n=1.

$T_n=5-3n$Tn=53n

1. Find $T_{30}$T30.

2. Find the first term less than $-150$150.

3. If $S_n$Sn is the sum of the first $n$n terms, find $S_{15}$S15.

4. Starting from $n=1$n=1, find the minimum number of terms required for the sum to be less than $-600$600.