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3.07 Further applications of growth and decay

Lesson

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.

a

What is the total charge for a 10 \text{ km} journey?

Worked Solution
Create a strategy

Add the pick-up fee to the number of kilometres travelled times the charge per kilometre.

Apply the idea
\displaystyle \text{Total charge}\displaystyle =\displaystyle 1.50+1.95\times 10Add the charges
\displaystyle =\displaystyle 1.50+19.50Evaluate the multiplication
\displaystyle =\displaystyle \$21.00Evaluate the addition
b

We want to describe this situation as a recursive sequence. To start with, state the initial condition T_0.

Worked Solution
Create a strategy

Use the initial pick-up fee.

Apply the idea

Initially the only charge is the pick-up fee of \$1.50.T_0=1.50

c

Write a recurrence relation for T_n in terms of T_{n-1} which defines the price of an n km trip.

Worked Solution
Create a strategy

Find how much each term increases by.

Apply the idea

We know that T_0=1.50. Let's find the next couple of terms in the sequence.

\displaystyle T_1\displaystyle =\displaystyle 1.50+1.95\times 1Add the charges
\displaystyle =\displaystyle 3.45Evaluate
\displaystyle T_2\displaystyle =\displaystyle 1.50+1.95\times 2Add the charges
\displaystyle =\displaystyle 5.40Evaluate

We can find the difference between the terms to find the common difference:

\displaystyle T_1-T_0\displaystyle =\displaystyle 3.45-1.5Subtract the first 2 terms
\displaystyle =\displaystyle 1.95Evaluate
\displaystyle T_2-T_1\displaystyle =\displaystyle 5.4-3.45Subtract the next 2 terms
\displaystyle =\displaystyle 1.95Evaluate

So the arithmetic sequence increases by 1.95 each time. This is because for every kilometre travelled, \$1.95 is added to the charge.

So the recurrence relation is given by:T_n=T_{n-1}+1.95,\, \, T_0=1.50

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.

a

What was the car purchased for in 2009?

Worked Solution
Create a strategy

Find how many years the card depreciated for.

Apply the idea

Since the car bought at the beginning of 2009, the value of the car decreased by \$50 each year for 6 years.

So, to find the purchase price of the car in 2009 we need to add 6 lots of \$50 to the value in 2015.

\displaystyle \text{Purchase price}\displaystyle =\displaystyle 1500+6\times 50Add 6 lots of \$50
\displaystyle =\displaystyle \$1800Evaluate
b

Plot the value of the car, V_n, on the graph from 2009 (represented by n=0) to 2015 (represented by n=6).

1
2
3
4
5
6
n
200
400
600
800
1000
1200
1400
1600
1800
V_n
Worked Solution
Create a strategy

Find the value of the car for each year, then plot the points with coordinates of the form (n,V_n).

Apply the idea

The value of the car in 2009 is \$1800. This means that it is represented by the point (0,1800).

Since the value of the car depreciates by \$50 each year, then this means the heights of the points as we move from left to right should decrease by \$50. Summarising the ordered pairs in a table, we have:

n0123456
V_n1800175017001650160015501500

So we get the following points:

1
2
3
4
5
6
n
200
400
600
800
1000
1200
1400
1600
1800
V_n
c

Write an explicit rule for the value of the car after n years. Give the rule in its expanded form.

Worked Solution
Create a strategy

Use the formula: V_n=V_1+\left(n-1\right)d.

Apply the idea

From part (b) we know that V_1=1750 and d=-50.

\displaystyle V_n\displaystyle =\displaystyle 1750+(n-1)\times (-50)Substitute the values
\displaystyle =\displaystyle 1750-50n+50Expand the brackets
\displaystyle =\displaystyle 1800-50nCombine like terms
d

Solve for the year n at the end of which the car will be worth half the price it was bought for.

Worked Solution
Create a strategy

Let V_n equal half the price from 2009, and use the rule from part (c) to solve for n.

Apply the idea

The car was originally bought for \$1800, Half of this price is \$900. So we let V_n=900.

\displaystyle V_n\displaystyle =\displaystyle 18000-50nWrite the rule
\displaystyle 1800-50n\displaystyle =\displaystyle 900Substitute V_n=900
\displaystyle 1800-50n-1800\displaystyle =\displaystyle 900-1800Subtract 1800 from both sides
\displaystyle -50n\displaystyle =\displaystyle -900Combine like terms
\displaystyle \dfrac{-50n}{-50}\displaystyle =\displaystyle \dfrac{-900}{-50}Divide both sides by -50
\displaystyle n\displaystyle =\displaystyle 18Evaluate

At the end of 18 years, the car will be worth half the price it was bought for.

Idea summary

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.

a

How many bacteria will there be in the body after 24 hours?

Worked Solution
Create a strategy

Use the formula: t_n=ar^{n}, where a=t_0.

Apply the idea

We are given the initial ammount of t_0=19 and since the bacteria doubles we know that r=2. Since it takes4 hours for the bacteria to double, n is equal to the number of 4-hour time periods in 24 hours, so n=\dfrac{24}{4}=6.

\displaystyle \text{Bacteria}\displaystyle =\displaystyle 19 \times 2^{6}Substitute the values
\displaystyle =\displaystyle 1216Evaluate
b

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.

Worked Solution
Create a strategy

Use the formula: t_n=ar^{n}, where a is the number of bacteria after the initial 24 hours.

Apply the idea

After the initial 24 hours there was 1216 bacteria which we will use as t_0 since this is when the vaccine is applied.

The vaccines kills \dfrac{1}{3} of the bacteria, but we want to find how much bacteria are left in the body. So we should use r=1-\dfrac{1}{3}=\dfrac{2}{3} to find the remaining bacteria.

Since it kills bacteria every 2 hours, n is equal to the number of 2-hour time periods in 24 hours, so n=\dfrac{24}{2}=12.

\displaystyle \text{Population}\displaystyle =\displaystyle 1216 \times \left(\dfrac{2}{3}\right)^{12}Substitute the values
\displaystyle \approx\displaystyle 9 \text{ bacteria}Evaluate and round
Idea summary

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}12345678910
\$2002252583003554265196407961000

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:

A 3 dimensional histogram of investments  during 10 months. Ask your teacher for more information.

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 relationExplicit equation
Arithmetic sequencet_{n+1}=t_n+d,\,t_1=at_n=a+d(n-1)
Geometric sequencet_{n+1}=r\times t_n,\,t_1=at_n=ar^{n-1}
First order linear recurrencet_{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.

a

How many weeds are in Uther’s garden at the end of the first month?

Worked Solution
Create a strategy

Increase the initial amount by the given percentage, and subtract the amount killed with herbicide.

Apply the idea

To increase an amount by 0.4\% we need to multiply by 1+0.004=1.004.

\displaystyle \text{Increased population}\displaystyle =\displaystyle 5000\times 1.004Increase 5000 by 0.4\%
\displaystyle =\displaystyle 5020Evaluate

Now we need to subtract the 750 killed to get:

\displaystyle W_1\displaystyle =\displaystyle 5020-750Subtract 750
\displaystyle =\displaystyle 4270Evaluate
b

Complete the recursive rule which describes this situation.W_{n+1}=⬚\times W_n-⬚,\, W_0=⬚

Worked Solution
Create a strategy

Consider what we multiplied by and subtracted in part (a).

Apply the idea

We are given the initial amount of W_0 =5000. In part (a) we multiplied W_0 by 1.004 and we subtracted 750 to find the next term. So the completed rule is:W_{n+1}=1.004\times W_n-750,\, W_0=5000

c

After how many months will Uther have a weed free garden?

Worked Solution
Create a strategy

Find the position of the first term less than or equal to zero.

Apply the idea

There are no weeds left when the number at the end of the month goes below zero. We can use our CAS calculator to find the first term less than or equal to zero in the sequence. We should find that this is the 7th term which means:\text{Number of months} = 7 \text{ months}

Reflect and check

We can check this by subtracting 7 lots of 750 from the initial amount of 5000.

\displaystyle 5000-7\times 750\displaystyle =\displaystyle -250Evaluate

Since we get an answer below zero, we know there will be no more weeds.

Idea summary

The first order linear recurrence relation is given by: t_{n+1}=k\times t_n+d,\,t_1=a

Outcomes

ACMGM070

use arithmetic sequences to model and analyse practical situations involving linear growth or decay; for example, analysing a simple interest loan or investment, calculating a taxi fare based on the flag fall and the charge per kilometre, or calculating the value of an office photocopier at the end of each year using the straight-line method or the unit cost method of depreciation

ACMGM074

use geometric sequences to model and analyse (numerically, or graphically only) practical problems involving geometric growth and decay; for example, analysing a compound interest loan or investment, the growth of a bacterial population that doubles in size each hour, the decreasing height of the bounce of a ball at each bounce; or calculating the value of office furniture at the end of each year using the declining (reducing) balance method to depreciate

ACMGM077

use first-order linear recurrence relations to model and analyse (numerically or graphically only) practical problems; for example, investigating the growth of a trout population in a lake recorded at the end of each year and where limited recreational fishing is permitted, or the amount owing on a reducing balance loan after each payment is made

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