Patterns in Numbers, 4, 6, 8, 10, 1, 4, 9, 16, 5, 1,, 4, 8, 16, 3, 1, 3, 9, 7, 81, Sequences Sequences A sequence is a list of real numbers:,,,,, with a rule for each 1,, 3,, 4, 6, 8, 10,,,,, 1, 4, 9, 16, 5,,,,, Fibonacci Sequence 1,, 4, 8, 16, 3,,,,,, 1, 3, 9, 7, 81,,,,, 3 Fibonacci Sequence in Nature 0, 1, 1,, 3, 5, 8, 13, 1, 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 1
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: 3 4 5 99 100 Gauss completed the task in seconds How? Gauss Genius 3 4 5 99 100 Gauss realised that addition of terms from opposite ends of the list always yielded an answer of 101: 1 3 4 5 99 100 100 99 98 97 96 1 101 101 101 101 101 101 101 100 101 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
Gauss Genius 3 4 5 99 100 5050 100 101 This gives us the general formula: 1 5050 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: 3 In our sample sequence 4,7,10,13,16, 4and we know 3 4 4 3 4 6 3 4 9 Arithmetic Series 3 3 1 Arithmetic Series 1 1 1 1 1 1 1 1 1 Arithmetic Series Overview The common difference is found by subtracting one term from the term which immediately follows it: To find the n th term in an arithmetic sequence, apply the following: To find the sum of the first n terms in an arithmetic sequence, apply the following: 1 3
Arithmetic Series Ex 1 Arithmetic Series Ex 1 In the following arithmetic series: 3,7,11,15, a) What is the 1 th term in this sequence? b) Find the sum of the first 1 terms in the sequence Solution: We can see that 3and 4 3 1 1 4 47 The 1 th term in the sequence is 47 b) Find the sum of the first 1 terms in the sequence 1 1 3 1 1 4 6 6 44 300 3 7 11 15 19 3 7 31 35 39 43 47 300 Arithmetic Series Ex In the following arithmetic series:,7,1,17, a) What is the 7 th term in this sequence? Solution: We can see that and 5 7 1 5 Arithmetic Series Ex b) Find the sum of the first 6 terms in the sequence 6 1 6 1 5 b) Find the sum of the first 6 terms in the sequence 13 The 7 th term is 13 13 4 15 1677 7 1 17 17 1677 Arithmetic Series Ex 3 Q A stack of telephone poles has 30 poles in the bottom row There are 9 poles in the second row, 8 in the next row, and so on How many poles are in the stack if there are 5 poles in the top row? Arithmetic Series Ex 3 Our arithmetic series is as follows: 5 6 7 9 30 We know that 5 and 1 If 5 is the 1 st term, 6 is the nd term, 7 is the 3 rd term then 30 must be the 6 th term Thus, to add up all the poles, we must find the sum of the first 6 terms ie 6 1 5 6 1 1 455 There are 455 poles in the stack 4
Arithmetic Series Ex 4 In an arithmetic sequence, the fifth term is 18 and the tenth term is 1 i Find the first term and the common difference ii Find the sum of the first fifteen terms of the sequence 418 9 1 418 91 530 6 Arithmetic Series Ex 4 418 4 618 4 First term is 4 and the common difference is 6 ii Find the sum of the first fifteen terms of the sequence 1 15 15 4 15 1 6 84 84 0 Geometric Sequences & Series A sequence is geometricif the ratio,, between any two consecutive terms is a constant This constant is called the common ratioand is usually denoted by Example:,4,8,16,3,64, The common ratio here is Geometric Sequences & Series Geometric Sequences & Series This can be shortened into the following equation: Geometric Sequences & Series - Overview To find the common ratio : To find the n th term in the geometric sequence, apply the following: To find the sum of the first n terms in a geometric sequence, apply the following: 5
Geometric Series Ex 1 Find the sum of the following geometric series: 5 10 0 40 1040 Solution: We know 5 and We need to find out how many terms are in the series 5 1040 048 log 0481 1 1 1 Geometric Series Ex 1 We now know that 1040 is the 1 th term so there are 1 terms in total in the series To find the sum of the geometric series we must find 5 1 5 4095 1 0475 The sum of the geometric series is 0,475 Geometric Series Ex Find the sum of the following geometric series: 3 1 48 3,07 Solution: We know 3 and 4 We need to find out how many terms are in the series 3 4 307 4 104 log 1041 51 6 Geometric Series Ex We now know that 307 is the 6 th term so there are 6 terms in total in the series To find the sum of the geometric series we must find 3 1 4 1 4 3 4095 3 4095 The sum of the geometric series is 4,095 GeometricSeries Ex 3 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 6 annual payments in the annual-payments option, with the first payment immediately, and the last payment in 5 years time The payments increase by 4% each year The amount advertised as the jackpot prize is the total amount of these 6 payments The cash-value option pays a smaller amount than this Geometric Series Ex 3 (a)if the amount of the first annual payment is, write down, in terms of, the amount of the second, third, fourth and 6th payments 1 st payment nd payment 104 3 rd payment 104 104 104 4 th payment 104 104 104 6 th payment 104 6
Geometric Series Ex 3 (b) The 6 payments form a geometric series Use this fact to express the advertised jackpot prize in terms of 104 104 104 104 1 104 1 104 1774 004 443 Geometric Series Ex 3 (c) Find, correct to the nearest dollar, the value of a that corresponds to an advertised jackpot prize of $1 5 million Jackpot prize 4431 1,500,000 4431 $485,196 This means that the initial payment will be $485,196and will increase by 4% with each subsequent payment Geometric Series Ex 4 Q In January 013, 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 013, HMV would pay 1 cent into the severance package for all staff on Feb 1 st, cent on Feb nd, 4 cent on Feb 3 rd, 8 cent on the 4 th, 16 cent on the 5 th 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? Geometric Series Ex 4 The payments would add up as follows: 001 00 004 008 016 We can see that this is a geometric series 001 There will be 8 payments as there are 8 days in February in 013, so we need to find the sum of the first 8 terms of this series Geometric Series Ex 4 Geometric Series Ex 4 In the end, HMV ended up paying out,684,35455 in severance fees to their Irish staff 001 1 001 1 68,435456 1,684,35455 7
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 questionsstarting 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: 5,10,0,40, We can see that it is a geometric sequence with 5and 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 5 560 The top prize on Dempsey s Double is,560 Sum to Infinity of Geometric Series lim lim This limit will exist if 1 The series will converge giving: 1 This is because 0 as if 1 Sum to Infinity of a Geometric Series lim lim This limit will not exist if 1 As such a series would diverge ie approach infinity This is because as if 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 1 Q Find the sum to infinity of the following series: 1 1 6 1 36 1 16 1 so the series will converge: 1 Sum to Infinity Ex Q Write the recurring decimal 0474747 as an infinite geometric series and hence as a fraction Solution: We can write the number as a geometric series: We know: 1 1 1 1 1 6 0474747 47 47 100 10,000 47 1,000,000 1 6 6 5 Now, we know and 8
Sum to Infinity Ex 1 so the series will converge and: Patterns 013 Pilot P1 Q9 47 100 1 1 1 100 1 47 99 47 100 0474747 47 99 100 99 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 Patterns (a) (i) Draw the fourth pattern (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 Patterns (b) Write an expression in nfor the number of triangles in the pattern of the sequence (c) Find an expression in n for the number of matchsticks needed to turn the 1 pattern into the pattern 9
Patterns (d) The number of matchsticks in the pattern in the sequence can be represented by the function where, Q and Find the value of and the value of Patterns (e) One of the patterns in the sequence has 4134 matchsticks How many small triangles are in that pattern? Patterns Ex Patterns Ex 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 ( ) are needed to make the staircase Change of changes is consistently constant Therefore the expression to represent how many total blocks ( ) are needed to make the staircase is QUADRATIC Patterns Ex Sketch a graph to represent this pattern Patterns Ex 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 and use the figures you know 1 3 63 3 1 43 9 3 6 10
Patterns Ex We can solve these simultaneous equations: 1 43 9 3 6 Eventually you should find Thus, can be written as 0 Patterns Ex How many blocks would be used to make a staircase that is 18 steps high? We know that represents the number of steps, so 18in this instance To find the number of blocks needed we use: 18 18 171 or 171 blocks would be used to make a staircase that is 18 steps high 11
Leaving Cert Helpdesk Number Sequences & Series UL 014 1 Write the recurring decimal 0474747 as an infinite geometric series and hence as a fraction By writing the recurring part as an infinite series, express the following number as a fraction of integers: 51111 3 In an arithmetic sequence, the fifth term is 18 and the tenth term is 1 (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 0 seats in the first row, in the second, 4 in the third, and so on, for 16 rows How many seats are there in the theatre?
Leaving Cert Helpdesk Number Sequences & Series UL 014 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) (ii) How far (up and down) will the ball have travelled when it hits the ground for the 6th time? Approximate the total distance that the ball will have travelled when it comes to rest 7 The winner of a $,000,000 sweepstakes will be paid $100,000 per year for 0 years The money earns 6% interest per year The present value of the winnings is 100,000 1 106 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 : 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 4 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 4 games in the coming season, how much would he earn through option? (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? (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?