IB Studies Level Mathematics Arithmetic Sequences and Series

The 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 100+97+94+91+… How much chocolate would you get altogether?

Arithmetic Series This series of numbers (100+97+94+…) is called an arithmetic series. We can easily solve this problem with the right Information and tools!

Arithmetic 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

Algebraically So for any term, n Tn = a +(n-1) d a a+ d a+2d a+3d ? T1 1st term T2 2nd term T3 3rd term T4 4th term Tn nth term a a+ d a+2d a+3d ? So for any term, n Tn = a +(n-1) d

Example Find the 20th term of the sequence 5,8,11,14,… Here a = 5 and d= 3 So Tn = 5+ (20-1)3 Tn = 62

So 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.

The 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?

The 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)

The Sum of an Arithmetic Series- Sn We can use l = a+ (n-1)d because l is the nth term of the series so substituting Sn = n/2 ((a + a+(n-1)d)) Which gives Sn = n/2 (2a + (n-1)d) Here’s a nice applet

Example using the summation formula Find the sum of the first 22 terms of the arithmetic series 6 + 4+ 2 +…. Using Sn = n/2 (2a + (n-1)d) Sn = 11 ( 12 +21 (-2)) Sn = -330

How many pieces of chocolate? Our series for the choc0late problem looks like this: 100+97+94+…..+ Here there are 34 terms n=34 since 100/3 So S34 = 34/2(200 +33(-3)) S34 = 1717 So you will eat 1717 pieces of chocolate after 34 days. How many pieces will you eat on the last day?

A nice summary of AP’s Here is a quick summary if you need it. Answers to sheet