Sequences revision

In a nutshell

General sequences are a list of numbers which can either be described by a term to term rule or position to term rule. These sequences have various patterns and may or may not have limits.

Defining a sequence

The term to term rule gives a term in a general sequence and the calculation to go to the next term. The position to term rule uses an equation to work out a number in a sequence if the position in the sequence is known.

Example 1

Given that $u_1 = 10$ and $u_{n+1} = 2u_n + 3$ , what is the $3^{rd}$ term in the sequence?

The number $n$ indicates the position in the sequence. So $u_1$ is the first term in the sequence. To find $u_2$ (the second term), substitute $n=1$ into the equation of the sequence:

$\begin{aligned} u_{1+1} &= 2u_1 +3 \\ u_2 &= 2(10) + 3 \end{aligned}$ 

 $u_2 = 23$ 

Substitute $n=2$ into the equation of the sequence to find the third term:

$\begin{aligned} u_{2+1} &= 2u_2 +3 \\ u_3 &= 2(23) + 3 \end{aligned}$

$\underline{u_3 = 49}$

Example 2

What is the $6^{th}$ term in a sequence with the formula $u_{2n} =\dfrac{ 4n +5}{2}$ ?

Find the value of $n$ which will give the $6^{th}$ term:

$\begin{aligned} u_{2n}&=u_6\\2n &= 6 \\ n&=3 \end{aligned}$

Substitute $n=3$ into the formula:

$u_{2(3)} = \dfrac {4(3) + 5 } {2}$

$\underline { u_6 = \dfrac {17}{2}= 8.5 }$

Increasing, decreasing or periodic sequences

Sequences may increase such that each term is always larger than the previous one, they may decrease such that each term is smaller than the previous one or they may be periodic such that the sequence eventually repeats itself.

Example 3

Find the first $4$ terms of $u_{n+1} = \dfrac{1}{u_n}$ given that $u_1 = 5$ . Describe the behaviour of this sequence.

Calculate $u_2, \ u_3,\ u_4$ :

$\begin{aligned} u_2 &= \dfrac{1}{5} \\ \\ u_3 &= \dfrac{1}{\frac15} = 5 \\ \\ u_4 &=\dfrac {1}{5} \end{aligned}$ 

Describe the behaviour of the sequence:

The sequence is periodic with a periodicity of $\underline2$ .

Note: Periodicity means how many terms there are that the sequence goes through before repeating.

Limits

A given general sequence may be convergent or divergent. A convergent sequence has a limit such that as $n \rightarrow \infin$ (" $n$ tends to infinity") the values of $u_n$ and $u_{n+1}$ will be the same. A divergent sequence has no limit, so as $n \rightarrow \infin$ the values of $u_n$ and $u_{n+1}$ will continue to be different.

Example 4

A convergent sequence is defined as $u_{n+1} = \dfrac35 u_n +4$ . What is the limit, $L$ ?

Consider $n \rightarrow \infin$ . You know that the sequence is convergent so there exists a limit $L$ such that

$u_{n+1} = u_n = L$

Using the sequence equation, solve for $L$ :

$L = \dfrac 35 (L) + 4$

$\dfrac 25 (L) = 4$

$2L = 20 \\ \underline{L = 10}$