number 1 - sequences & series...

1

Patterns in Numbers

2, 4, 6, 8, 10, …

1, 4, 9, 16, 25,…

1, �2, 4, �8, 16, �32, …

1, 3, 9, 27, 81,…

Sequences

A sequence is a list of real numbers:

�, �, �, �, … , �

with a rule � for each � � 1, 2, 3,…

Sequences

2, 4, 6, 8, 10,… �, �, �, �, �…

� � 2�________________________________

1, 4, 9, 16, 25, … �, �, �, �, �…

� � ��

1,�2, 4,�8, 16,�32,… �, �, �, �, �, �…

� � �2 ��

__________________________________

1, 3, 9, 27, 81,… �, �, �, �, �…

� � 3 ��

Fibonacci Sequence

0, 1, 1, 2, 3, 5, 8, 13, 21, 34,...

Sequence used in:

• Music to structure rhymes and rhythm• Computer games: Metal Gear Solid 4• Poetry• DNA Construction• Solar System arrangement.• Nature• Art: used by Marilyn Manson

Fibonacci Sequence in Nature

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Milky Way and Fibonacci Sequence

Apple Symbol and Fibonacci Sequence

Gauss’ Genius

• Carl Friedrich Gauss’ Primary school teacher wanted to keep the class busy while he completed other work so he asked the pupils to add all whole numbers between 1 and 100:

1 � 2 � 3 � 4 � 5 �…� 99 � 100

• Gauss completed the task in seconds. How?

Gauss’ Genius

1 � 2 � 3 � 4 � 5 �…� 99 � 100

Gauss realised that addition of terms from opposite ends of the list always yielded an answer of 101:

1 � 2 � 3 � 4 � 5 �…� 99 � 100100 � 99 � 98 � 97 � 96 �…� 2 � 1

101 � 101 � 101 � 101 � 101 �…� 101 � 101

100 1012 � 5050

Series

Given a sequence �, �, �, �, �… we can construct the corresponding series:

��

��

�� ...�� …� �

Series

�� …� �

�� can be written in shorthand:

��

� �

This can be interpreted as the sum starting from ! � 1 and stopping when ! � �

3

Gauss’ Genius

1 � 2 � 3 � 4 � 5 �…� 99 � 100 � 5050

�"�##

$ �� 100 1012 � 5050

This gives us the general formula:

�"�

$ �� 12

Arithmetic Series

An arithmetic sequence is one where consecutive terms have a common difference:

% � �&� � �

Example: Given the arithmetic sequence:

4, 7, 10, 13, 16, …

Here, the common difference is 3. So % � 3.

Arithmetic Series

In an arithmetic sequence, the first term ' �( is usually called a so:

� � ) � � ) � % � � ) � 2% � � ) � 3%

In our sample sequence 4, 7, 10, 13, 16,… ) � 4 and we know % � 3

� � ) � 4 � � ) � % � 4 � 3 � � ) � 2% � 4 � 6 � � ) � 3% � 4 � 9

Arithmetic Series

� � ) � � ) � % � � ) � 2% � � ) � 3%... � � ) � '� � 1(%

�� …� �

�� ) � ) � % � ) � 2% � ) � 3% �…� *) � � � 1 %+

Arithmetic Series

�� ) � ) � % � ) � 2% �⋯� *) � � � 2 %+ � *) � � � 1 %+

�� ) � � � 1 % � ) � � � 2 % �⋯� ) � 2% � ) � % � )

2�� 2) � � � 1 % � 2) � � � 1 % � 2) � � � 1 % �…� 2) � � � 1 % � 2) � � � 1 %

2�� 2) � � � 1 %

�� 2 2) � � � 1 %

Arithmetic Series – Overview

The common difference % is found by subtracting one term from the term which immediately follows it:

% � �&� � �

To find the nth term in an arithmetic sequence, apply the following:

� � ) � '� � 1(%

To find the sum of the first n terms in an arithmetic sequence, apply the following:

�� 2 2) � � � 1 %

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Arithmetic Series – Ex. 1

In the following arithmetic series:

3, 7, 11, 15, …

a) What is the 12th term in this sequence?

b) Find the sum of the first 12 terms in the sequence.

Solution:

We can see that ) � 3 and % � 4

� � ) � '� � 1(%

�� 3 � 12 � 1 4

�� 47

The 12th term in the sequence is 47.



�� 2 2) � � � 1 %

�� 122 2'3( � 12 � 1 4

�� 6 6 � 44 � 300

3 � 7 � 11 � 15 � 19 � 23 � 27 � 31 � 35 � 39 � 43 � 47� 300


In the following arithmetic series:

2, 7, 12, 17, …

a) What is the 27th term in

this sequence?

b) Find the sum of the first

26 terms in the sequence.

Solution:

We can see that ) � 2 and % � 5

� � ) � '� � 1(%

�- � 2 � 27 � 1 5

�- � 132

The 27th term is 132.



�� 2 2) � � � 1 %

�� 262 2'2( � 26 � 1 5

�� 13 4 � 125 � 1677

2 � 7 � 12 � 17…� 127 � 1677


Q. A stack of telephone poles has 30 poles in the bottom row. There are 29 poles in the second row, 28

in the next row, and so on. How many poles are in the stack if there are 5 poles in the top row?


Our arithmetic series is as follows:

5 � 6 � 7 �…� 29 � 30

We know that ) � 5 and % � 1

If 5 is the 1st term, 6 is the 2nd term, 7 is the 3rd term then 30 must be the 26th

term.

Thus, to add up all the poles, we must find the sum of the first 26 terms i.e. ��

�� 2 2) � � � 1 %

�� 262 2'5( � 26 � 1 1

�� 455There are 455 poles in the stack.

5


In an arithmetic sequence, the fifth term is �18and

the tenth term is 12.

i. Find the first term and the common difference.

ii. Find the sum of the

first fifteen terms of the sequence.

� � ) � '� � 1(%

� � ) � 4% � �18 �# � ) � 9% � 12

) � 4% � �18�) � 9% � �12

�5% � �30

% � 6


) � 4% � �18

) � 4'6( � �18

) � �42

First term is �42 and the common difference is 6

ii. Find the sum of the first fifteen terms of the sequence.

�� 2 2) � � � 1 %

�� 152 2'�42( � 15 � 1 6

�� 152 �84 � 84 � 0

Geometric Sequences & Series

A sequence is geometric if the ratio, ./01./ , between any

two consecutive terms is a constant.

This constant is called the common ratio and is usually denoted by ".

Example:2, 4, 8, 16, 32, 64, …

The common ratio here is 2.


� � ) � � )" � � )"� � � )"�... � � )"��

�� …� �

�� ) � )" � )"� � )"� �…� )"��


�� ) � )" � )"� � )"� �…� )"��

This can be shortened into the following equation:

�� )1 � "�1 � "

Geometric Sequences & Series -

OverviewTo find the common ratio ":

" � �&� �

To find the nth term in the geometric sequence, apply the following:

� � )"��

To find the sum of the first n terms in a geometric sequence, apply the following:

�� )1 � "�1 � "

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Geometric Series – Ex. 1

Find the sum of the following geometric series:

5 � 10 � 20 � 40 �⋯� 10240

Solution:

We know ) � 5 and " � 2

We need to find out how many terms are in the series.

� � )"��

5 2�� 10240

2�� 2048

log� 2048 � � � 1

11 � � � 1

� � 12


We now know that 10240 is the 12th term so there are 12 terms in total in the series.

To find the sum of the geometric series we must find ��

�� )1 � "�1 � "

�� 51 � 2��1 � 2

�� 5 �4095�1

�� 20475

The sum of the geometric series is 20,475


Find the sum of the following geometric series:

3 � 12 � 48 �…� 3,072

Solution:

We know ) � 3 and " � 4

We need to find out how many terms are in the series.

� � )"��

3 4�� 3072

4�� 1024

log� 1024 � � � 1

5 � � � 1

� � 6


We now know that 3072 is the 6th term so there are 6 terms in total in the series.

To find the sum of the geometric series we must find ��

�� )1 � "�1 � "

�� 31 � 4�1 � 4

�� 3 �4095�3

�� 4095

The sum of the geometric series is 4,095


Most lottery games in the USA allow winners of

the jackpot prize to choose between two forms

of the prize: an annual-payments option or

a cash-value option.

In the case of the New York Lotto, there are 26 annual

payments in the annual-payments option, with the first payment immediately, and the last payment in 25 years time.

The payments increase by 4% each year. The amount

advertised as the jackpot prize is the total amount of these 26 payments. The cash-value option pays a smaller amount than

this.


(a) If the amount of the first annual payment is ), write down, in terms of ), the amount of the second, third, fourth and 26th payments.

1st payment � )2nd payment� 1.04)3rd payment � 1.04 1.04) � 1.04 �)4th payment � 1.04 1.04 �) � 1.04 �)...26th payment � 1.04 ��)

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(b) The 26 payments form a geometric series. Use this fact to express the advertised jackpot prize in terms of ).

) � 1.04) � 1.04 �) � 1.04 �) �…� 1.04 ��)

�� )1 � "�1 � "

�� )1 � 1.04��1 � 1.04 � ) �1.7724�0.04

�� 44.312) � 5)6789:8"!;<


(c) Find, correct to the nearest dollar, the value of a that corresponds to an advertised jackpot prize of $21·5 million.

Jackpot prize � 44.312)

21,500,000 � 44.312)

) � $485,196

This means that the initial payment will be $485,196 and will increase by 4% with each subsequent payment.


Q. In January 2013, HMV staff in Limerick City held a sit-in to aid negotiations for a fair severance package. The company were offering a total of €800,000 to be shared out amongst all the HMV staff in Ireland as a severance package.

The staff were not happy with this figure and instead suggested that, in the month of February 2013, HMV would pay 1 cent into the severance package for all staff on Feb. 1st, 2 cent on Feb. 2nd, 4 cent on Feb. 3rd, 8 cent on the 4th, 16 cent on the 5th and so on for the rest of that month. HMV agreed to the deal immediately. How much did HMV end up paying to the staff?


The payments would add up as follows:

€0.01 � €0.02 � €0.04 � €0.08 � €0.16…

We can see that this is a geometric series.

) � 0.01 " � 2

There will be 28 payments as there are 28 days in February in 2013, so we need to find the sum of the first 28 terms of this series.


�� )1 � "�1 � "

��? � 0.011 � 2�?1 � 2

��? � 0.011 � 268,435456

�1

��? � 2,684,354.55


In the end, HMV ended up paying out €2,684,354.55 in severance fees to their Irish staff.

8

Dempsey’s Double

The Ian Dempsey Breakfast Show on Today FM runs a daily competition called Dempsey’s Double. Every morning one lucky listener will have the chance to try their hand at winning thousands of euro:

“We ask you ten questions starting with €5 for the first correct answer. For every answer you get right after that, we double your cash.”

What’s the top prize in this competition?

Dempsey’s Double

The sequence is:5, 10, 20, 40, …

We can see that it is a geometric sequence with ) � 5 and " � 2.

If we find the tenth term of this sequence then we will find the prize money if they answer all of the ten questions correctly.

� � )"��

�# � 5 2 �#�� 2560

The top prize on Dempsey’s Double is €2,560

Sum to Infinity of Geometric Series

�@ � � � � � � �…� � �…

�@ � lim�→@ �� lim�→@ )1 � "�1 � "

This limit will exist if " D 1 The series will converge giving:

�@ � )1

1 � "

This is because "� → 0 as � → ∞ if " D 1

Sum to Infinity of a Geometric Series

�@ � lim�→@ �� lim�→@ )1 � "�1 � "

This limit will not exist if " F 1 As such a series would diverge i.e. approach infinity.

This is because "� → ∞ as � → ∞ if " F 1 .

So, to find the sum to infinity of a geometric series, the modulus of the common ratio "must be less than 1.

Sum to Infinity – Ex. 1Q. Find the sum to infinity of the following series:

1 � 16 �136 �

1216 �…

We know:) � 1

" � 16

" D 1 so the series will converge:

�@ � ) 11� "

�@ � 1 11 � 16

�@ � 65

Sum to Infinity – Ex. 2

Q. Write the recurring decimal 0.474747… as an infinite geometric series and hence as a fraction.

Solution: We can write the number as a geometric series:

0.474747… � 47100 �

4710,000 �

471,000,000 �…

Now, we know ) � �-�## and " � �

�##

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Sum to Infinity – Ex. 2

" D 1 so the series will converge and:

�@ � )1

1 � "

�@ �47100

11 � 1

100� 47100

10099

�@ �4799

0.474747… � 4799

Patterns 2013 Pilot P1 Q9

Shapes in the form of small equilateral triangles can be made using matchsticks of equal length. These shapes can be put together into patterns. The beginning of a sequence of these patterns is shown below

Patterns

(a) (i) Draw the fourth pattern

Patterns

(ii) The table below shows the number of small triangles in each pattern and the number of matchsticks needed to create each pattern. Complete the table.

Patterns

(b) Write an expression in n for the number of triangles in the

�GH pattern of the sequence.

Patterns

(c) Find an expression in n for the number of matchsticks

needed to turn the � � 1 GH pattern into the �GH pattern.

10

Patterns

(d) The number of matchsticks in the �GH pattern in the sequence can be represented by the function I� � )�� J�where ), J ∈ ℚ and � ∈ M. Find the value of ) and the value of J

Patterns

(e) One of the patterns in the sequence has 4134 matchsticks. How many small triangles are in that pattern?

Patterns – Ex. 2

The diagram is the side view of a staircase that is 4 steps high. This staircase is made by stacking 10 blocks as shown.

How many blocks would be used to make a staircase that is 18 steps high?

If �represents the number of steps in a staircase, write an expression to represent how many total blocks (N) are needed to make the staircase.

Patterns – Ex. 2

Change of changes is consistently constant ... Therefore the expression to represent how many total blocks (N) are needed to make the staircase is QUADRATIC

Patterns – Ex. 2

Sketch a graph to represent this pattern

Patterns – Ex. 2

Write an expression to represent how many total blocks (B) are needed to make the staircase:

You might be able to figure this out in your head but if you’re struggling then recognise that this will be a quadratic in the form N � )�� J� � 6and use the figures you know

1 � )'1(��J'1( � 6 ⇒ ) � J � 6 � 1

3 � )'2(��J'2( � 6 ⇒ 4) � 2J � 6 � 3

6 � )'3(��J'3( � 6 ⇒ 9) � 3J � 6 � 6

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Patterns – Ex. 2

We can solve these simultaneous equations:

) � J � 6 � 14) � 2J � 6 � 39) � 3J � 6 � 6

Eventually you should find ) � �� J � �

� 6 � 0

Thus, N � )�� J� � 6 can be written as

N � ��

�� or N � PQ&P

�

Patterns – Ex. 2

How many blocks would be used to make a staircase that is 18 steps high?

We know that � represents the number of steps, so � � 18 in this instance. To find the number of blocks needed we use:

N � �� 2

N � '18(��182 � 171

171 blocks would be used to make a staircase that is 18 steps high.

Leaving Cert Helpdesk Number – Sequences & Series UL 2014

1. Write the recurring decimal 0.474747… as an infinite geometric series and hence as a fraction.

2. By writing the recurring part as an infinite series, express the following number as a fraction of integers: 5.21212121…

3. In an arithmetic sequence, the fifth term is �18 and the tenth term is 12.

(i) Find the first term and the common difference. (ii) Find the sum of the first fifteen terms of the sequence.

4. Legend tells us that the game of chess was invented hundreds of years ago by the Grand Vizier Sissa Ben Dahir for King Shirham of India. The King loved the game so much that he offered his Grand Vizier any reward he wanted for having created the game. “Majesty, give me one grain of wheat to go on the first square of the chess board, two grains to place on the second square, four grains to place on the third square, eight on the fourth square and so on until all the squares on the board are covered”. Note: there are 64 squares on a chess board. How many grains of wheat did the King owe Sissa Ben Dahir through this deal?

5. Cinema theatres are often built with more seats per row as the rows move towards the back.

In screen 3 of the local cinema there are 20 seats in the first row, 22 in the second, 24 in the third, and so on, for 16 rows. How many seats are there in the theatre?

Leaving Cert Helpdesk Number – Sequences & Series UL 2014

6. A superball, or bouncy ball as it sometime called, dropped from the top of St. John’s Cathedral in Limerick (308 ft high) rebounds three-quarters of the distance fallen. (i) How far (up and down) will the ball have travelled when it hits the ground for

the 6th time?

(ii) Approximate the total distance that the ball will have travelled when it comes to rest.

7. The winner of a $2,000,000 sweepstakes will be paid $100,000 per year for 20 years. The money earns 6% interest per year. The present value of the winnings is

�100,000 � 11.06�

� �

��

Compute the present value and interpret its meaning.

8. Stoke City midfielder Glenn Whelan’s contract has just expired. His agent informs him that the club want him to sign a new contract and have given him two options:

• Option 1: A 1-year contract with no bonuses through which he is guaranteed to earn €1,400,000 over the course of the contract.

• Option 2: A 1-year contract which will pay him based only on the number of competitive games he appears in (as either a sub or starter) – he will get €35,000 for the first game, €38,000 for the second, €41,000 for the third game, €44,000 for the fourth game and so on. In other words, he will have €3,000 added to his payment each time he plays a game.

(a) Glenn Whelan appeared in 24 games in the most recent season and thinks he will appear in about the same number of games in the coming season. If he plays in 24 games in the coming season, how much would he earn through option 2?

(b) Stoke City typically play 44 competitive games per season (this includes Premier

League and domestic cup competitions). If Stoke City play 44 games in the coming season, what is the maximum amount of money Glenn Whelan can earn through option 2?

(c) How many games would Glenn Whelan need to play in to earn more than

€1,400,000 over the course of the 1-year contract if he were to pick option 2?

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