2017year%10% general%mathematics% …

2017 Year 10 General Mathematics

Topic 4: Number Patterns and Recursion This topic includes:

• the concept of a sequence as a function

• use of a first-‐order linear recurrence relation to generate the terms of a number sequence

• tabular and graphical display of sequences.

Key knowledge

• the concept of sequence as a function and its recursive specification

• the use of a first-‐order linear recurrence relation to generate the terms of a number sequence including the special cases of arithmetic and geometric sequences; and the rule for the nth term, tn, of an arithmetic sequence and a geometric sequence and their evaluation

• the use of a first-‐order linear recurrence relation to model linear growth and decay, including the rule for evaluating the term after n periods of linear growth or decay

• the use of a first-‐order linear recurrence relation to model geometric growth and decay, including the use of the rule for evaluating the term after n periods of geometric growth or decay

Key skills

• use a given recurrence relation to generate an arithmetic or a geometric sequence, deduce the rule for the nth term from the recursion relation and evaluate

• use a recurrence relation to model and analyse practical situations involving discrete linear and geometric growth or decay

• formulate the recurrence relation to generate the Fibonacci sequence and use this sequence to model and analyse practical situations.

For this topic ALL QUESTIONS are included in these notes at the end of each section.

More resources available at

http://drweiser.weebly.com

GENERAL MATHEMATICS 2017

1. Sequences A sequence is a list of numbers in a particular order. The numbers or items in a sequence are

called the terms of the sequence. They may be generated randomly or by a rule.

Randomly generated sequences

Recording the numbers obtained while tossing a die would give a randomly generated sequence, such as:

3, 1, 2, 2, 6, 4, 3, ...

Because there is no pattern in the sequence there is no way of predicting the next term.

Consequently, random sequences are of no relevance to this topic and will NOT be considered.

Rule based Sequences

Writing down odd numbers starting at 1 would result in a sequence generated by a rule:

1, 3, 5, 7, 9, 11, 13, ...

There is a rule that allows us to state the next term in the sequence.

‘add 2 to the current odd number’

For example:

to find the term after 13, just add 2 to 13, to get 13 + 2 = 15.

The group of three dots (… ) at the end of the sequence is called an ellipsis. An ellipsis is used to show that the sequence continues. In this topic, we will look at sequences that can be generated by a rule.

Naming the terms in a sequence

The symbols 𝑉0, 𝑉1, 𝑉2, are used as labels or names for the first, second and third terms in the sequence. In the labels 𝑉0, 𝑉1, 𝑉2 the numbers 0, 1, 2 are called subscripts. The subscripts tell us the position of each term in the sequence. So, 𝑉10 is just a name for the term in the sequence NOT the value of the term.

Term 1 2 3 4 5 6

n n=0 n=1 n=2 n=3 n=4 n=5

Vn V0 V1 V2 V3 V4 V5 e.g. 1, 3, 5, 7, 9, 11

Term 1 2 3 4 5 6

n n=0 n=1 n=2 n=3 n=4 n=5

Vn V0=1 V1=3 V2=5 V3=7 V4=9 V5=11

NUMBER PATTERNS AND RECURSION

Arithmetic sequences

Sequences that are generated by adding or subtracting a fixed amount to the previous term are called arithmetic sequences.

The fixed amount we add or subtract to form an arithmetic sequence recursively is called the common difference. The symbol d is often used to represent the common difference.

• If a sequence is known to be arithmetic, the common difference can be calculated by simply subtracting any pair of successive terms.

• If a sequence is not known to be arithmetic BUT is found to have a common difference then the sequence is arithmetic.

Common Difference, d

In an arithmetic sequence, the fixed number added to (or subtracted from) each term to make the next term is called the common difference, where:

𝑑 = 𝑎𝑛𝑦 𝑡𝑒𝑟𝑚 − 𝑡ℎ𝑒 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠 𝑡𝑒𝑟𝑚 𝑑 = 𝑉8 − 𝑉9 𝑑 = 𝑉: − 𝑉8 𝑑 = 𝑉; − 𝑉: 𝑎𝑛𝑑 𝑠𝑜 𝑜𝑛,

For example, the common difference for the arithmetic sequence 20, 25, 30, …is:

𝑑 = 𝑉8 − 𝑉9 = 25 − 20 = 5 or 𝑑 = 𝑉: − 𝑉8 = 30 − 25 = 5 etc…

Example 1 Finding the common difference in an arithmetic sequenceFind the common difference for the following arithmetic sequences and use it to find the 3rd term in the sequence: a) 2,5,8,...

b) 25,23,21,...


Using repeated addition on a CAS calculator to generate a sequence

As we have seen, a recursive rule based on repeated addition, such as ‘to find the next term, add 6’, is a quick and easy way of generating the next few terms of a sequence. However, it becomes tedious to do by hand if we want to find, say, the next 20 terms. Fortunately, your CAS calculator can semi-‐automate the process.

Graphs of arithmetic sequences

If we plot the values of the terms of an arithmetic sequence (Vn) against their number (n) or position in the sequence, we will find that the points lie on a straight line. An upward slope indicates regular growth and a downward slope reveals decay at a constant rate. A line with positive slope rises from left to right. A negative slope falls from left to right.


Exercise 1.

1. Find the required terms from the sequence: 6, 11, 16, 21, 26, 31. . .

a) V1 b) V3 c) V4 d) V5 e) V2 f) V0

2. For each sequence state the value of the named terms: i) V1 ii) V3 iii) V0

a) 6, 10, 14, 18, ...

b) 2, 8, 32, 128, ...

c) 29, 22, 15, 8, ...

d) 96, 48, 24, 12, ...

3. Find out which of the sequences below is arithmetic. Give the common difference for each sequence that is arithmetic.

a) 8, 11, 14, 17, ... b) 7, 15, 22, 30, ... c) 11, 7, 3, −1, ...

d) 12, 9, 6, 3, ... e) 16,8,4,2, ... f) 1, 1, 1, 1, ...

4. For each of these arithmetic sequences, find the common difference and the 5th term.

a) 5, 11, 17, 23, ... b) 17, 13, 9, 5, ... c) 11, 15, 19, 23, ...

d) 8, 4, 0, −4, ... e) 35, 30, 25, 20, ... f) 1.5, 2, 2.5, 3, ...

5. Give the next two terms in each of these arithmetic sequences.

a) 17, 23, 29, 35, ... b) 14, 11, 8, 5, ... c) 2, 1.5, 1.0, 0.5, ...

d) 27, 35, 43, 51, ... e) 33, 21, 9, −3, ... f) 0.8, 1.1, 1.4, 1.7, ...


6. Using your CAS calculator: a) Generate the first six terms of the arithmetic sequence: 1, 6, 11...and write down V5.

b) Generate the first 12 terms of the arithmetic sequence: 45, 43, 41... and write down V12.

c) Generate the first 10 terms of the arithmetic sequence: 15, 14, 13, ... and write down V10.

d) Generate the first 15 terms of the arithmetic sequence: 0, 3, 6, ... and write down V15.

7. The number of sticks used to make the hexagonal patterns opposite form the arithmetic sequence: 6, 11, 16, . . .

a) Write the common difference for this sequence.

b) Using your CAS calculator, determine the number of matches needed to form:

i) pattern 6 ii) pattern10

8. After one week of business Fumbles Restaurant had 320 wine glasses.After two weeks, they only had 305 wine glasses. On average 15 glasses are broken each week.Use your CAS calculator, to determine how many weeks it takes at that breakage rate for there to be only 200 glasses left?

9. Elizabeth stored 350 songs on her phone in the first month. In each month that followed she stored 35 more songs.Using your CAS calculator:

a) determine the number of songs she had stored after each of the first 4 months

b) determine the number of songs she had stored by the end of the first year.


10. a) Graphing the terms of the arithmetic sequence 4, 7, 10, . . . i. Construct a table showing the term number (n) and its value (tn) for the first five terms in the

sequence. ii. Use the table to plot the graph.

iii. Describe the graph.

b) Graphing the terms of the arithmetic sequence 9, 7, 5, . . . i. Construct a table showing the term number (n) and its value (tn) for the first five terms in the

sequence. ii.Use the table to plot the graph.

iii.Describe the graph.


2. Using a Recurrence Relation to generate and analyse an arithmetic sequence Generating the terms of a first-‐order recurrence relations

A first-‐order recurrence relation relates a term in a sequence to the previous term in the same sequence. To generate the terms in the sequence, only the initial term is required. A recurrence relation is a mathematical rule that we can use to generate a sequence. It has two parts:

1. a starting point: the value of one of the terms in the sequence 2. a rule that can be used to generate successive terms in the sequence.

For example, in words, a recursion rule that can be used to generate the sequence: 10, 15, 20, ...can be written as follows:

1. Start with 10. 2. To obtain the next term, add 5 to the current term and repeat the process.

A more compact way of communicating this information is to translate this rule into symbolic form. We do this by defining a subscripted variable. Here we will use the variable Vn, but the V can be replaced by any letter of the alphabet.

Let Vn be the term in the sequence after n iterations*. Using this definition, we now proceed to translate our rule written in words into a mathematical rule.

Starting value (n=0) Rule for generating the next term Recurrence relation

(two parts: starting value plus rule)

V0=10

Vn+1=Vn+5 Next term =current term +5

V0=10 Vn+1=Vn+5 Starting value rule

Note: Because of the way we defined Vn, the starting value of n is 0. At the start there have been no applications of the rule. This is the most appropriate starting point for financial modelling.

Example 2 For the sequence 2, 7, 12, 17, …

a) Determine if it is an arithmetic sequence 𝑑 = 𝑎𝑛𝑦 𝑡𝑒𝑟𝑚 − 𝑡ℎ𝑒 𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠 𝑡𝑒𝑟𝑚 Yes it is arithmetic 𝑑 = 𝑉8 − 𝑉9 = 7 − 2 = 5 𝑑 = 𝑉: − 𝑉8 = 12 − 7 = 5

b) Hence, if it is an arithmetic sequence, state the common difference

hence 𝑑 = 5

c) State the Recurrence Relation for the sequence Recurrence relation 𝑉9 = 2, 𝑉?@8 = 𝑉? + 𝑑, here 𝑑 = 5 so: 𝑇ℎ𝑒 𝑅𝑒𝑐𝑢𝑟𝑟𝑒𝑛𝑐𝑒 𝑅𝑒𝑙𝑎𝑡𝑖𝑜𝑛 𝑖𝑠: 𝑉9 = 2, 𝑉?@8 = 𝑉? + 5

d) Using your CAS list the first 10 terms of the sequence (hint 𝑛 = 0 → 𝑛 = 9)

*Each time we apply the rule it is called an iteration.


The importance of the Starting Term

In the example 2 above, If the same rule is used with a different starting point, it will generate different sets of numbers. Example 2 𝑉I = 2, 𝑉?@8 = 𝑉? + 5 The first five terms were: 2, 7, 12, 17, 22

If V0 =1 then, they would be: 1, 6, 11, 16, 21 If V0 =3 then, they would be: 3, 8, 13, 18, 23

Here you can clearly see that the effect the value of the starting point has. Hence, a recurrence relation MUST have it’s starting value stated at ALL TIMES

Finding other Terms in a recurrence relation (A General Rule) We can also use recurrence relations to find previous terms, but we need two pieces of information

1. The rule, in terms of Vn+1 and Vn 2. The term number and its value. i.e. n=2 and V2=10 (note if n=0, 1, 2, … then n=2 is the 3rd term)

Finding the 𝑛th term in an arithmetic sequence

In Example 2(a), above, the sequence is 4, 7, 10, 13, … where 𝑉9 = 4 and 𝑑 = 3. Writing this out gives:

𝑉9 = 4 = 𝑉9 + 𝟎×3 = 4 𝑉𝟏 = 𝑉9 + 3 = 𝑉9 + 3 = 𝑉9 + 𝟏×3 = 7 𝑉𝟐 = 𝑉8 + 3 = 𝑉9 + 3 + 3 = 𝑉9 + 𝟐×3 = 10 𝑉𝟑 = 𝑉: + 3 = 𝑉9 + 3 + 3 + 3 = 𝑉9 + 𝟑×3 = 13 𝑉𝟒 = 𝑉; + 3 = 𝑉9 + 3 + 3 + 3 + 3 = 𝑉9 + 𝟒×3 = 16

We can see that a pattern has emerged, that is:

𝑉? = 𝑉9 + 𝑛×𝑑, where 𝑉? the 𝑛th term,

𝑉9 is the starting term, 𝑑 = common difference 𝑛 = position number of the term.

Example 3- Finding the nth term of an arithmetic sequence

a) Find t5, the 5th term in the arithmetic sequence: 21, 18, 15, 12, . . .

b) Find t10, the 10th term in the arithmetic sequence: 9, 7, 5, …


Exercise 2.

1. a) Generate and graph the first five terms of the sequence defined by the recurrence relation: 𝑉0

= 15, 𝑉?@8

= 𝑉?

+ 5 where 𝑛 ≥ 1. b) Calculate the value of the 45th term in the sequence.

2. a) Generate and graph the first five terms of the sequence defined by the recurrence relation: 𝑉0

= 60, 𝑉?@8

= 𝑉?

− 5 where 𝑛 ≥ 1. b) Calculate the value of the 10th term in the sequence.

3. a) Generate and graph the first five terms of the sequence defined by the recurrence relation: 𝑉9=15, 𝑉?@8 = 𝑉𝑛

+ 35 where 𝑛 ≥ 1.

b) Calculate the value of the 15th term in the sequence.


4. The Llama shapes have been made using blocks.

Llama 0 Llama 1 Llama 2

Let 𝐵𝑛 be the number of blocks used to make the 𝑛th Llama shape.The number of blocks used to make each Llama shape is generated by the recurrence relation:

𝐵9

= 7, 𝐵?@8

= 𝐵?

+ 4 a) Count and record the number of blocks used to make the first, second and third Llama shapes.

b) Use the recurrence relation for Bn to generate the first five terms of the sequence of perimeters for these shapes.

c) Use a rule to calculate the number of blocks needed to make the Llama 8 shape. 5. The BBQ shapes have been made using blocks, each with a side length of 1 unit.

BBQ 0 BBQ 1 BBQ 2

The perimeter of each BBQ shape can be found by counting the sides of the blocks around the outside of the shape.Let Pn be the perimeter of the nth BBQ shape.The perimeters for this sequence of BBQ shapes is generated by the recurrence relation:

𝑃9

= 16, 𝑃?@8

= 𝑃𝑛

+ 6 a) Count and record the perimeters of the first, second and third BBQ shapes.

b) Use the recurrence relation for Pn to generate the first four terms of this sequence of perimeters.

c) Draw the fourth BBQ shape, find its perimeter and check if the recurrence relation correctly predicted the perimeter.

d) Use the rule for the nth term for this sequence to predict the perimeter of the 10th BBQ shape (𝑃89).


3. Geometric sequences The common ratio, r

In a geometric sequence, each new term is made by multiplying the previous term by a fixed number called the common ratio, r. This repeating or recurring process is another example of a sequence generated by recursion. In the sequence:

each new term is made by multiplying the previous term by 3. The common ratio is 3. In the sequence:

64 32 16 8 4

each new term is made by halving the previous term. In this sequence, we are multiplying each term by ×8

:, which is equivalent to dividing by 2. The common ratio is 8

:. New terms in a geometric sequence

𝑉9, 𝑉8, 𝑉;, 𝑉W, … are made by multiplying the previous term by the common ratio, 𝑟.

Common Ratio, 𝒓 In a geometric sequence, the common ratio, 𝑟, is found by dividing the next term by the current term.

𝐶𝑜𝑚𝑚𝑜𝑛 𝑅𝑎𝑡𝑖𝑜, 𝑟 =𝑐𝑢𝑟𝑟𝑒𝑛𝑡 𝑡𝑒𝑟𝑚𝑝𝑟𝑒𝑣𝑖𝑜𝑢𝑠 𝑡𝑒𝑟𝑚 =

𝑉8𝑉9=𝑉:𝑉8=𝑉;𝑉:= ⋯

Note: we will only consider values of 𝑟 > 0 (consider what happens if 𝑟 < 0)

Example 4 Find the common ratio in each of the following geometric sequences.

a) 3, 12, 48, 192, ...

b) 81, 27, 9, 3, …


Identifying geometric sequences

To identify a sequence as a geometric sequence, it is necessary to find the ratio between multiple pairs of successive terms. If they are common (the same), then it is a geometric sequence.

Example 5 Which of the following sequences are geometric sequences?

a) 2, 10, 50, 250, ...

b) 3, 6, 18, 36, …

Using repeated multiplication on a CAS calculator to generate a geometric sequence

As we have seen, using a recursive rule based on repeated multiplication, such as ‘to find the next term, multiply by 2’, is a quick and easy way of generating the next few terms of an geometric sequence. It would be tedious to find the next 50 terms. Fortunately, your CAS calculator can semi-‐automate the process of performing multiple repeated multiplications and do this very quickly.


Graphs of geometric sequences

In contrast with the straight-‐line graph of an arithmetic sequence, the values of a geometric sequence lie along a curve. Graphing the values of a sequence is a valuable tool for understanding applications involving growth and decay.

In the graph above the sequence 2, 4, 8, 16, 32, 64, 128, 256, … is an example of geometric growth where 𝒓 = 𝟐. In the graph above the sequence 256, 128, 64, 32, 16, 8, 4, 2, … is an example of geometric decay where 𝒓 = 𝟏

𝟐

Graphs of Geometric Sequences

Graphs of Geometric Sequences (for 𝑟 > 0, i. e. 𝑟 𝑖𝑠 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒)

§ increasing when 𝑟 is greater than 1, 𝑟 > 0

§ decreasing towards zero when 𝑟 is between 0 and 1, 0 < 𝑟 < 1.

Exercise 3

1. Find out which of the following sequences are geometric. Give the common ratio for each sequence that is geometric. a) 4, 8, 16, 32, ... b) 1, 3, 9, 27, … c) 5, 10, 15, 20, … d) 5, 15, 45, 135, …

e) 24, 12, 6, 3, ... f) 3, 6, 12, 18, … g) 4, 8, 12, 16, … h) 2, 4, 8, 16…


2. Find the missing terms in each of these geometric sequences. a) 7, 14, 28, __, __, ... b) 3, 15, 75, __, __, … c) 4, 12, __, __, 324, …

d) __, __, 20, 40, 80, … e) 2, __, 32, 128, __, ... f) 3, __, 27, __, 243, 729, …

3. Use your graphics calculator to generate each sequence and find 𝑉c, the sixth term.

a) 7, 35, 175, ... b) 3, 18, 108, … c) 96, 48, 24, …

d) 4, 28, 196, … e) 160, 80, 40, ... f) 11, 99, 891, …

4. Consider each of the geometric sequences below.

i. Find the next two terms. ii. Show the terms in a graph. iii. Describe the graph.

a) 3, 6, 12, ...

b) 8, 4, 2, ...


4. Using a recurrence relation to generate and analyse a geometric sequence Consider the geometric sequence below: 2, 6, 18, ... We can continue to generate the terms of this sequence by recognising that it uses the rule: ‘to find the next term multiply the current term by 3 and keep repeating the process’. A recurrence relation is a way of expressing this rule in a precise mathematical language. The recurrence relation that generates that sequence 2, 6, 18, . . . is:

𝑉0

= 2, 𝑉?@8 = 3×𝑉𝑛 The rule tells us that:‘the first term is 2, and each subsequent term is equal to the current term multiplied by 3’. Understanding this, we proceed to generate the sequence term-‐by-‐term as follows:

𝑉9 = 𝟐 𝑉8 = 𝑉9×3 = 𝟐×3 = 𝟔 𝑉: = 𝑉8×3 = 𝟔×3 = 𝟏𝟖 𝑉; = 𝑉:×3 = 𝟏𝟖×3 = 𝟓𝟒 𝑉W = 𝑉;×3 = 𝟓𝟒×3 = 𝟏𝟔𝟐 and so on

The recurrence relation for generating a geometric sequence is: 𝑡ℎ𝑒 𝑠𝑡𝑎𝑟𝑡𝑖𝑛𝑔 𝑡𝑒𝑟𝑚 𝑉9, 𝑉?@8 = 𝑉?×𝑟,

where 𝑉? the 𝑛th term, 𝑉9 is the starting term,

𝑟 = common ratio 𝑛 = position number of the term.

Example 6 Generate the recurrence relation for the following geometric sequences

a) 4, 8, 16, 32, ...

b) 1, 3, 9, 27, … c) 5, 10, 15, 20, …


General form of the recurrence relation for a Geometric Sequence

Considering the sequence above: 2, 6, 18, ... 𝑉9 = 𝟐 𝑉8 = 𝑉9×3 = 𝑉9×3 = 𝑉9×3𝟏 = 𝟔 𝑉𝟐 = 𝑉8×3 = 𝑉9×3×3 = 𝑉9×3𝟐 = 𝟏𝟖 𝑉𝟑 = 𝑉:×3 = 𝑉9×3×3×3 = 𝑉9×3𝟑 = 𝟓𝟒 𝑉𝟒 = 𝑉;×3 = 𝑉9×3×3×3×3 = 𝑉9×3𝟒 = 𝟏𝟔𝟐 and so on

The nth term of geometric sequence can be found by the recurrence relation: 𝑡ℎ𝑒 𝑠𝑡𝑎𝑟𝑡𝑖𝑛𝑔 𝑡𝑒𝑟𝑚 𝑉9, 𝑉? = 𝑉9×𝑟?,

where 𝑉? the 𝑛th term, 𝑉9 is the starting term,

𝑟 = common ratio 𝑛 = position number of the term.

Example 7 a) Generate the first 5 terms of the sequence defined by the recurrence relation: 𝑉9 = 5, 𝑉?@8 = 2×𝑉𝑛

b) Graph the first 5 terms c) Write down a general recurrence rule to calculate the value of the nth term in the sequence and

use it to find 𝑉89.


Exercise 4.

1. a) Generate first five terms of the geometric sequence defined by the recurrence relation: 𝑡0

= 1000, 𝑡𝑛

+

1

= 1.1𝑡𝑛.

b) Write down a general recurrence rule to calculate the value of the nth term in the sequence and use it to find 13th term in the sequence correct to two decimal places. 2. a) Generate the first five terms of the geometric sequence defined by the recurrence relation:

𝑡0

= 256, 𝑡?@8

= 0.5𝑡𝑛.

b) Write down a general recurrence rule to calculate the value of the nth term in the sequence and use it to find 10th term in the sequence. 3. a) Generate the first five terms of the geometric sequence defined by the recurrence relation:

𝑡1

= 10 000, 𝑡?@8

= 1.25𝑡?. Give values to the nearest whole number.

b) Write down a general recurrence rule to calculate the value of the nth term in the sequence and use it to find 25th term in the sequence.


4. A sheet of paper is in the shape of a rectangle. When the sheet is folded once and opened, 2 rectangles are formed either side of the crease. When a sheet is folded twice and opened, 4 rectangles are created, and so on.

Note: in the above diagram, 𝑛 = 0 and hence 𝐹9 are not shown because that is just the unfolded paper

Let 𝐹𝑛 be the number of rectangles created by n folds.The sequence for the number of rectangles created is generated by the recurrence relation:

𝐹0

= 1, 𝐹?@8 = 2𝐹𝑛 a) Use the recurrence relation for 𝐹𝑛 to generate the first five terms of the sequence.

b) Write down a general recurrence relation for the 𝑛th term in the sequence and use it to calculate the number of rectangles after 5 𝑎𝑛𝑑 10 folds.

c) Using your calculator, generate the terms of the sequence to check your answer to b).

6. As a park ranger, Megan has been working on a project to increase the number of rare native orchids in Wilsons Promontory National Park. At the start of the project, a survey found 200 of the orchids in the park. It is assumed from similar projects that the number of orchids will increase by about 18% each year. a) State the first term 𝑉9, and the common ratio 𝑟, for the geometric sequence

for the number of orchids each year.

b) Find a rule for the number of orchids at the start of the 𝑛th year.

c) How many orchids are predicted in 10 years’ time?


Chapter Summary Sequence A sequence is a list of numbers in a particular order.

Arithmetic sequence

In an arithmetic sequence, each new term is made by adding afixed number, called the common difference, d, to the previous term.

Example: 3, 5, 7, 9, . . . is made by adding 2 to each term.

The common difference, d, is found by taking any term and subtracting its previous term, e.g. V1 – V0.In our example above, d = 5 − 3 = 2.

Recurrence relation for an arithmetic sequence

A recurrence relation for an arithmetic sequence has the form

𝑉0 = 𝑎, 𝑉?@8 = 𝑉𝑛 + 𝑑

where d = common difference and a = first term. In our example:

𝑉0

= 3, 𝑉?@8 = 𝑉?

+ 2

General Rule for finding Vn, the nth term in an arithmetic sequence:

𝑉𝑛

= 𝑉9 + 𝑛×𝑑

To find 𝑉? in our example: put 𝑛 = 10, 𝑎 = 3, 𝑑 = 2

𝑉? = 3 + 10×2 = 23

The graph of an arithmetic sequence: " values lie along a straight line

• Increasing values when d>0 (positive slope) • Decreasing values when d<0 (negative slope)

Linear growth & decay

An arithmetic sequence can be used to model linear growth (d > 0) or linear decay (d < 0).

Geometric sequence

In a geometric sequence, each term is made by multiplying the previous term by a fixed number, called the common ratio, 𝑟. Example: 5, 20, 80, 320, . .. is made by multiplying each term by 4.

The common ratio, r, is found by dividing any term by its previous term,

e.g. 𝑟 = nonp= nq

no= nr

nq…

In our example: 𝑟 = nonp= :9

c= 4

Recurrence relation for a geometric sequence

Recurrence relation for a geometric sequence:

𝑉9

= 𝑎, 𝑉?@8 = 𝑟×𝑉? where r = common ratio and a = first term. In our example:

𝑉9

= 5, 𝑉?@8 = 4×𝑉? General Rule for finding 𝑡𝑛, the 𝑛th term, in a geometric sequence:

𝑉? = 𝑉9×𝑟?where 𝑉9 =first term and 𝑟 = common ratio.

To find 𝑉s in our example: put 𝑛 = 6, 𝑎 = 5, 𝑟 = 4 into: 𝑉? = 5×4s = 20,480 The graph of a geometric sequence:

Values increase when 𝑟 > 1 Values decrease towards zero when 0 < 𝑟 < 1


Table of Contents Topic 4: Number Patterns and Recursion ............................................................................................... 1

This topic includes: ................................................................................................................................. 1 Key knowledge ....................................................................................................................................... 1 Key skills ................................................................................................................................................. 1

1. Sequences .......................................................................................................................................... 2

Randomly generated sequences ............................................................................................................ 2 Rule based Sequences ............................................................................................................................ 2 Naming the terms in a sequence ............................................................................................................ 2 Arithmetic sequences ............................................................................................................................. 3 Example 1 ............................................................................................................................................... 3 Using repeated addition on a CAS calculator to generate a sequence .................................................. 4 Graphs of arithmetic sequences ............................................................................................................ 4 Exercise 1. .............................................................................................................................................. 5

2. Using a Recurrence Relation to generate and analyse an arithmetic sequence ................................ 8

Generating the terms of a first-‐order recurrence relations ................................................................... 8 Example 2 ............................................................................................................................................... 8 The importance of the Starting Term ..................................................................................................... 9 Finding other Terms in a recurrence relation (A General Rule) ............................................................. 9 Finding the nth term in an arithmetic sequence .................................................................................... 9 Example 3-‐ Finding the nth term of an arithmetic sequence ................................................................. 9 Exercise 2. ............................................................................................................................................ 10

3. Geometric sequences ....................................................................................................................... 12

The common ratio, r ............................................................................................................................. 12 Example 4 ............................................................................................................................................. 12 Identifying geometric sequences ......................................................................................................... 13 Example 5 ............................................................................................................................................. 13 Using repeated multiplication on a CAS calculator to generate a geometric sequence ...................... 13 Graphs of geometric sequences ........................................................................................................... 14 Exercise 3 ............................................................................................................................................. 14

4. Using a recurrence relation to generate and analyse a geometric sequence .................................. 16

Example 6 ............................................................................................................................................. 16 General form of the recurrence relation for a Geometric Sequence ................................................... 17 Example 7 ............................................................................................................................................. 17 Exercise 4. ............................................................................................................................................ 18

Table of Contents ................................................................................................................................. 21


Modelling practical situations (linear growth and decay) Linear growth and decay is commonly found around the world. They occur when a quantity increases or decreases by the same amount at regular intervals. Everyday examples include the paying of simple interest or the depreciation of the value of a new car by a constant amount each year.

An example of linear growth is the investment of money, such as putting it in a savings account where the sum increases over time.

An example of linear decay is the money owned to repay a loan, the sum of money owned will decrease over time.

Example 4: Jelena puts $5000 into an investment that earns simple interest at a rate of $50 per month. (a) Set up a recurrence relation that represents Jelena’s situation as an arithmetic sequence, where Vn+1

is the amount in Jelena’s account after n months. (b) Use your equation from part (a) to determine the amount in Jelena’s account at the end of each of

the first 6 months.

n Vn d Vn+1=Vn+d

n = 0 V1 =

n = 1 V2 =

n = 2 V3 =

n = 3 V4 =

n = 4 V5 =

n = 5 V6 =

(c) Calculate the amount in Jelena’s account at the end of 18 months

n = 18, V18 =


Depreciating assets Many items, such as electronic equipment, depreciate over time because of wear and tear. Unit cost depreciation is a way of calculating the value of depreciation according to its use. For example, the value of a cars depreciation is based on how many kilometres it has driven. The value of an item at any given time can be calculated and is referred to as its future value. The write-off value or scrap value of an asset is the point at which the asset is effectively worthless, that is when the value is equal to $0 due to depreciation.

Example 5: Loni purchases a new car for $25000 and decides to depreciate it at a rate of $0.20 per km. (a) Set up an equation to determine the value of the car after n km of use. (b) Use your equation from part (a) to determine the future value of the car after it has 7500km on its

clock.

2017year%10% general%mathematics% …

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