Chapter
Recursive Formulas and
Recurrence Relations
4
(FM2 2010M101)
[13 marks]
X.
The charge, in dollars, for a single trip on a tollway depends on the number of sections of road
that a motorist travels and the type of toll pass that the motorist uses.
(a) Using toll pass A, the charge for travelling along n sections of road in a single trip on the
tollway is given by the nth term of the following arithmetic sequence.
$4.50, $6.20, $7.90 ...
(i)
Show that the common difference for this sequence is $1.70.
[1]
,
(ii) Find the charge for travelling along five sections of road in a single trip on the
tollway using toll pass A.
[1]
(iii) One motorist paid $16.40 for a single trip on the tollway using toll pass A. How
many sections of road did this motorist travel along?
[1]
(iv) At one entry point, fifteen motorists entered the tollway. The first motorist travelled
along one section of road. The second motorist travelled along two sections of road.
The third motorist travelled along three sections of road and so on. Find the total
amount of money that these 15 motorists paid for their trips, assuming they all used
toll pass A.
[2]
(v) Using toll pass A, the charge, in dollars, An, for travelling along n sections of road in
a single trip on the tollway is given by the equation
An+i = mAn + k
Ai = 4.50
Write the values of m and k in the boxes below.
m=
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45
[2]
Creelman Exam Questions: Mathematics Applications ATAR Course Units 3 and 4, 2017
1. (cont)
(b) Different charges apply when motorists use toll pass B. With toll pass B, the charge, in
dollars, Bn, for travelling along n sections of road in a single trip on the tollway is given
by the equation
Bn+i = 0.9 Bn + 3
(i)
B] = 5
Explain the meaning of B3 = 5 in terms of the context of this problem.
[1]
(ii) Find B3, the charge for travelling along three sections of road in a single trip using
toll pass B.
[1]
(iii) This difference equation indicates that there is a maximum charge which motorists
who use toll pass B may pay. What is this maximum charge?
[2]
(c)
A motorist wishes to get the best value for money when travelling on the tollway.
Compare the charges for a single trip using toll pass A and toll pass B. Explain when it
would be better for the motorist to use each pass.
[2]
46
Chapter 4: Recursive Formulas and Recurrence Relations
2.
(FM2 20UM102)
[6 marks]
Streaming Media is a company that provided Internet access to its customers. Customers are
charged for data transfer during each calendar month (Jan, Feb,. ..) as follows.
• The first gigabyte (GB) of data transfer costs $4.
• Each GB of data transfer after this costs 20 cents less than the previous one.
• Once the cost of a GB of data transfer reaches zero, any additional data transfer during that
month is also free.
(a) How much does the fifth GB of data transfer cost?
[1]
(b) What is the cost of transferring a total of 8GB of data in a calendar month?
[1]
.W
A customer paid the maximum charge for data transfer in a month.
(c) What is the minimum number of GB of data that this customer could have transferred?
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47
[1]
Creelman Exam Questions: Mathematics Applications ATAR Course Units 3 and 4, 2017
2. (cont)
(d) Let Dn be the number of gigabytes (GB) that Danny transfers in his nth month with
Streaming Media. Danny finds that his data transfer each month can be determined
using the difference equation
Dn+i = 1.5Dn - 1 where D j = 6
(i)
How many GB of data does Danny transfer in his third month?
[1]
(ii) Determine how much Danny will be charged in his third month.
[i]
f-
(e)
According to the difference equation above, in which month will Danny's transfer first
exceed 100GB of data?
[1]
48
Chapter 4: Recursive Formulas and Recurrence Relations
(FM2 2013M104)
[3 marks]
Apple trees are growing in an orchard. Over time, some of the trees stop producing enough
apples and are removed at the end of the year in which this first occurs. Immediately
afterwards, a fixed number of new apple trees will be planted.
The total number of apple trees growing in the orchard at the end of the nth year, An,
immediately after the planting of the new apple trees for that year, is modelled by the
difference equation
An+1 = 0.8An + k
A1 = 18 000
where k is the number of new apple trees planted at the end of each year.
(a) What percentage of apple trees will be removed at the end of each year?
[1]
(b) Assume 100 new apple trees are planted at the end of each year. Determine how many
apple trees will be growing in the orchard at the end of the third year, immediately after
the planting of the new apple trees for that year.
[1]
(c) Determine the number of new apple trees, k, that needs to be planted at the end of each
year so that there will always be 18 000 apple trees growing in the orchard.
[1]
49
Creelman Exam Questions: Mathematics Applications ATAR Course Units 3 and 4, 2017
4.
(3ABA4AT 2013:CF0l)
[4 marks]
A recursive sequence is defined by un = pun-1 + q. Given that iq = -8, u2 = 8 and u3 = 4, write
down two equations and solve simultaneously to determine the values of p and q.
50
Chapter 4: Recursive Formulas and Recurrence Relations
5.
[6 marks]
(APP SAT:CF03)
The set of numbers 1, 3, 6,10 is named triangular because the units (objects) can be arranged
to form triangles, as in the diagram below.
(a)
Draw the next pattern in the diagram below and give its value.
(c) Give the first order recurrence for the sequence.
[2]
[3]
Creelman Exam Questions: Mathematics Applications ATAR Course Units 3 and 4, 2017
6.
(APP SAT:CF04)
[9 marks]
A wedding photographer is quoting the following price for producing a wedding album for
the newlyweds:
A fixed minimum cost of $150, with 80 photos in a hard-backed album. Further photos may
also be added in lots of 10 photos at $0.70 per photo, up to a maximum of 200 photos.
He wants to set up a table below, showing:
• the type of album where T^ is the basic album, with 80 photos at a cost of $150
• the number of photos in each of the possible album sizes
• the cost in dollars of each of the different albums.
(a) Complete each of the blank cells of the table.
Type
Ti
Number of
pictures
80
$ cost of
album
$150
t2
t3
[3]
t4
Ts
T
1n
200
.w
(b)
Write a rule that will calculate the number of pictures in album type = Tn.
[3]
(c)
Write a rule that will calculate the cost of album type = cn.
[3]
52
Chapter 4: Recursive Formulas and Recurrence Relations
7.
(a)
(APP SAT.CA05)
[3 marks]
Complete the table below for the first five terms of the sequence, defined by
an+1 = an + (n + 1), a1 = 1
n
1
2
3
4
[2]
5
an
(b)
Evaluate a50.
[1]
53
Creelman Exam Questions: Mathematics Applications ATAR Course Units 3 and 4, 2017
8.
(APP SAT:CA06)
[12 marks]
A fish farm operates a fish breeding pond in which the population of a particular fish
increases by 3% per month.
(a)
Give the recurrence formula to calculate the fish population at the end of each month,
assuming the rate does not change and the initial population is 1000 fish.
[2]
(b)
Calculate the population numbers at the end of the first six months of its operation,
given the initial population is 1000 fish.
[2]
(c)
After the initial six months, 40 fish per month are removed at the end of each month.
Assuming the population growth is maintained at 3%, how many fish are expected to be
in the tank at the end of 12 months?
[3]
(d) Describe what is happening to the population.
(e)
[1]
Estimate, to the nearest whole number, the maximum number of fish that may be
removed from the tank per month without the numbers of fish decreasing.
[4]
54
Chapter 4: Recursive Formulas and Recurrence Relations
9.
(APP SAT:CA07)
[9 marks]
For the recurrence relation an+i = an + 0.6 and a0 = 3.1
(a)
Deduce the rule for the nth term of the relation.
(b)
Check the truth of the following proposition
(c)
Prove the above proposition can be generalised to an+2 = 2 x an+i - a.t
55
[3]
= 2 xa9-a8
[2]
[4]
Creelman Exam Questions: Mathematics Applications ATAR Course Units 3 and 4, 2017
10.
[12 marks]
(APP 2016:CF4)
(a) Given the sequence 256,128, 64, 32, ...
(i)
Write a recursive rule for the sequence
(ii) Deduce a rule for the nth term of this sequence. Hence, calculate the 15th term,
leaving your answer as a fraction.
[2]
[3]
(b) Use the recursive definitions given to state the first three terms of each of the following
sequences.
Tn + 7, Ti = 11
[2]
(ii) Tn+1 = !-5Tn, T2 = 7.5
[2]
(i)
Tn+1 =
(c) Consider the sequence 12, 7, 2, -3, ...
By deducing a rule for the nth term, or otherwise, determine which term of the sequence
is -168 .
[3]
56