Ideas
Applications of arithmetic sequences
We have seen many applications of linear growth or decay when studying linear functions previously. Hence, arithmetic sequences apply in many areas of life, including simple interest earnings, straight-line depreciation, monthly rental accumulation and many others. For example, if you are saving money in equal instalments, the cumulative savings at each savings period form an arithmetic sequence. If you are travelling down a highway at a constant speed, the amount of petrol left in the tank, if measured every minute of the trip, forms another arithmetic progression. In fact any time you notice a quantity changing in equal amounts at set time periods, then you can consider that process as being arithmetic.
Depreciation refers to the situation where items or investments lose value over time. We will consider two types of depreciation in this course:
Straight line depreciation - the value reduces by the same amount every time period. This is an application of arithmetic sequences.
Reducing balance depreciation - the value reduces by a percentage of the previous value every time period. This is an application of geometric sequences.
This application will be examined in further detail in the financial applications section of this course.
When we describe a recursive rule we know that it requires two parts. Firstly the rule describing how the sequence recurs and secondly, the initial condition, describing where to start. In previous chapters we have mainly used t_1 for the initial condition, referring to the first term of the sequence. However, sometimes it can be useful to use t_0 meaning the initial term. t_0 is particularly useful in financial questions and population questions where we start with an initial amount and then look for the amount in the days/weeks/month after that starting point. For example, consider a situation where we start with \$100 and each week this amount increases by \$20. If we want to know how much we have after 5 weeks and we use the initial condition of t_0=100 then 5 weeks later is t_5 where as if we used the initial condition of t_1=100 then 5 weeks later is t_6, which can be a bit confusing. Both would give the same answer but using t_0 in this case makes the term number we are looking for match the number of weeks.
Examples
Example 1
Zuber is a taxi service that charges a \$1.50 pick-up fee and \$1.95 per kilometre of travel.
What is the total charge for a 10 \text{ km} journey?
We want to describe this situation as a recursive sequence. To start with, state the initial condition T_0.
Write a recurrence relation for T_n in terms of T_{n-1} which defines the price of an n km trip.
Example 2
A car bought at the beginning of 2009 is worth \$1500 at the beginning of 2015. The value of the car has depreciated by a constant amount of \$50 each year since it was purchased.
What was the car purchased for in 2009?
Plot the value of the car, V_n, on the graph from 2009 (represented by n=0) to 2015 (represented by n=6).
Write an explicit rule for the value of the car after n years. Give the rule in its expanded form.
Solve for the year n at the end of which the car will be worth half the price it was bought for.
Depreciation refers to the situation where items or investments lose value over time. Straight line depreciation - the value reduces by the same amount every time period. This is an application of arithmetic sequences.
Sometimes it can be useful to use t_0 to mean the initial term, particularly in financial questions and population questions where we start with an initial amount and then look for the amount in the days/weeks/month after that starting point.
Applications of geometric sequences
It is also very common to encounter applications of geometric sequences such as compound interest, exponential growth of bacteria, exponential decay of radioactive elements. If we have an amount increasing or decreasing by a constant factor at set time periods, then you can consider that process as being geometric.
Examples
Example 3
To test the effectiveness of a new vaccine, a certain bacteria is introduced to a body and the number of bacteria is monitored. Initially, there are 19 bacteria in the body, and after four hours the number is found to double.
How many bacteria will there be in the body after 24 hours?
The vaccine is applied after 24 hours, and is found to kill one third of the germs every two hours. How many bacteria will there be left in the body 24 hours after applying the vaccine? Assume the bacteria stops multiplying and round your answer to the nearest integer if necessary.
It is very common to encounter applications of geometric sequences such as compound interest, exponential growth of bacteria, exponential decay of radioactive elements.
Applications of first order linear recurrence
The graph of a recurrence relation can provide a visual understanding of the way the sequence behaves. For instance it may show whether the terms are always increasing or always decreasing in size. It might also show that the terms change linearly, or as a quadratic curve, or as some other known form, and this might provide a hint as to a possible explicit form of the sequence.
Consider for example a stockbroker's claim that \$200 can be turned into a \$1000 in 10 months. He claims that he can make 30\% a month on the investment, for the cost of \$35 a month.
Such a claim seems incredulous, but given the percentage earnings and the monthly fee stated, a recurrence relation can be formed as: T_{n+1}=1.3T_n-35,\,T_1=200
Note that the coefficient 1.3 is equivalent to 130\%, which is a 30\% increase on the previous month's amount as shown by the formula. The subtraction of \$35 represents the monthly fee.
We can tabulate the size of the investment at the beginning of the month for the first 10 months as follows, using month numbers and the amounts rounded to whole dollars:
| \text{Month} | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|---|
| \$ | 200 | 225 | 258 | 300 | 355 | 426 | 519 | 640 | 796 | 1000 |
The beginning of month 2 shows that the investment has grown to \$225, determined using the recurrence formula, so that T_2=1.3\times 200-35=225. Similarly, the beginning of month 3 is determined as T_3=1.3\times 225-35=257.50 which, rounded to whole dollars, becomes \$258.
The sequence of 10 months can be graphed using a 3-dimensional histogram as follows:
Note the increase in the original investment, with the rate of change in value increasing as well. Although not strictly geometric (the \$35 deduction affects the ratio between successive terms), the trend is certainly apparent - the higher the amount, the faster it grows. This type of growth is typical of recurrence relationships that have the form T_{n+1}=k\times T_n\pm d with k>1, and is an example of the first order linear recurrence relations we investigated in our last lesson.
Here is the table for recurrence relations and their explicit equations.
| Recurrence relation | Explicit equation | |
|---|---|---|
| Arithmetic sequence | t_{n+1}=t_n+d,\,t_1=a | t_n=a+d(n-1) |
| Geometric sequence | t_{n+1}=r\times t_n,\,t_1=a | t_n=ar^{n-1} |
| First order linear recurrence | t_{n+1}=k\times t_n+d,\,t_1=a |
Examples
Example 4
Uther’s garden has 5000 weeds in it whose population is increasing at a rate of 0.4\% per month. At the end of each month Uther kills 750 weeds with herbicide.
How many weeds are in Uther’s garden at the end of the first month?
Complete the recursive rule which describes this situation.W_{n+1}=⬚\times W_n-⬚,\, W_0=⬚
After how many months will Uther have a weed free garden?
The first order linear recurrence relation is given by: t_{n+1}=k\times t_n+d,\,t_1=a