Modelling with sequences and series

In a nutshell

A real life model can be represented by a sequence whose series can be taken to find the total amount of something usually over a given time period.

Arithmetic series

Many scenarios can be represented by an arithmetic sequence and hence calculations can be performed using an arithmetic series.

Example 1

Derek's business makes $£30,000$ in profit in the first year. He predicts that his profits will increase by $£10,000$ each year. How long will it take him to make $£1,000,000$ ? State why the model which has been given may not be suitable.

Write out some terms of this sequence:

$30000, 40000,50000,60000...$ 

This is an arithmetic sequence, identify the first term and common difference:

$a= 30000 \\ d= 10000$

Create a model to represent this sequence:

$\begin{aligned} u_n &= a+(n-1)d \\u_n &= 30000 + (n-1)(10000) \end{aligned}$ 

This arithmetic sequence gives the predicted profit made by Derek's business in year $n$ .

Calculate how many years it will take to make $£1,000,000$ :

$S_n = \dfrac {n}2 (2a + (n-1)d)$ 

$1000000= \dfrac {n}2 (2(30000)+(n-1)(10000) )$ 

$2000000= n (60000 + 10000n-10000)$ 

$2000000= n (50000 + 10000n)$ 

$2000000= 50000n + 10000n^2 \\ 200= 5n + n^2 \\ 0=n^2+5n-200$ 

$n = \dfrac{-5 \pm \sqrt{825}}{2}$ 

$n$ is the number of years so it will take on positive values:

$n=\dfrac {-5+ \sqrt{825}}{2}=11.861406...$ 

$\underline{ n=11.9 \ \textit{years}\ (1 \ d.p)}$ 

State a reason why this model may not be valid:

It is unlikely that Derek's profits will increase by the same amount each year. Business growth is more likely to be proportional to its size.

Geometric series

Other scenarios can be represented by a geometric sequence allowing calculations to be performed using a geometric series.

Example 2

Each time a piece of paper is folder its thickness doubles. The thickness of a given piece of paper is $0.5mm$ . How thick will the paper be after $15$ folds. State why this model will have limitations.

Identify some terms of this sequence:

$0.5 , 1, 2, 4$ 

There is a common ratio between each term and the first term is given:

$\begin{aligned} r&=2 \\ a&=0.5\end{aligned}$ 

Create a model to represent this sequence:

$u_n = 0.5(2)^{n-1}$ 

Find the thickness after $15$ folds:

$u_{15} = 0.5(2)^{15-1} = 0.5(2)^{14}$ 

$\underline{U_{15} = 8192 \text{ mm} =8.192\text{ m}}$

State the limitations of this model:

It is not possible to fold a piece of paper indefinitely as is suggested by this model.