Presentation on theme: "IB Studies Level Mathematics"— Presentation transcript:
1IB Studies Level Mathematics Arithmetic Sequences and Series
2The Family Block of Chocolate Imagine if I gave you a family block of chocolate which was made up of 100 small squares every day.However as each day passed I started to eat a few pieces before I gave it to you.On the second day I eat three pieces so you get 97 pieces.On the third day I eat six pieces, on the fourth I eat nine pieces!So the amount of chocolate you get every day is …How much chocolate would you get altogether?
3Arithmetic SeriesThis series of numbers ( …) is called an arithmetic series.We can easily solve this problem with the right Information and tools!
4Arithmetic Sequences and series Have a look at the following sequences:An Arithmetic series is a series of numbers in which each term is obtained from the previous term by adding or subtracting a constant.The constant we add or subtract each time is called the common difference, “d” In our chocolate example this was -3. The first term is called “a” (here it was 100).The letter “n” is used to denote the number of terms
5Algebraically So for any term, n Tn = a +(n-1) d a a+ d a+2d a+3d ? T1 1sttermT22nd termT33rd termT44th termTnnth termaa+ da+2da+3d?So for any term, nTn = a +(n-1) d
6ExampleFind the 20th term of the sequence5,8,11,14,…Here a = 5 and d= 3So Tn = 5+ (20-1)3Tn = 62
7So what about our chocolate problem? We need to know how to sum an arithmetic sequence in order to solve our chocolate problem.Here’s a neat proof to show you the formula.
8The Sum of an Arithmetic Series- Sn Let the last term of an Arithmetic series be l.Sn= a + a+d + a+2d + a+3d + … + l-d + l Eqn (1)Now re-writing this backwards! Yes backwards!Sn = l + l-d + l-2d + l-3d + …. +a+d + a Eqn (2)We are now going to add the two equations together- can you see why? What cancels out?
9The Sum of an Arithmetic Series- Sn So 2Sn = lots of (a+l) but how many if there are “n” terms?Yes there are n lots of (a+l)This gives 2Sn = n(a+l)But what if we don’t know the last term?Sn = n/2 (a+l)
10The Sum of an Arithmetic Series- Sn We can use l = a+ (n-1)d because l is the nth term of the series so substitutingSn = n/2 ((a + a+(n-1)d))Which givesSn = n/2 (2a + (n-1)d)Here’s a nice applet
11Example using the summation formula Find the sum of the first 22 terms of the arithmetic series ….Using Sn = n/2 (2a + (n-1)d)Sn = 11 ( (-2))Sn = -330
12How many pieces of chocolate? Our series for the choc0late problem looks like this: …..+Here there are 34 terms n=34 since 100/3So S34 = 34/2( (-3))S34 = 1717So you will eat 1717 pieces of chocolate after 34 days.How many pieces will you eat on the last day?
13A nice summary of AP’s Here is a quick summary if you need it. Answers to sheet