1. IB Studies Level Mathematics
Arithmetic Sequences and Series

2. 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 …
How much chocolate would you get altogether?

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

4. 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

5. 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

6. 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

7. 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.

8. 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?

9. 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)

10. 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

11. Example 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

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

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