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IB Studies Level IB Studies Level Mathematics Mathematics

Arithmetic Sequences Arithmetic Sequences and Seriesand Series

The Family Block of ChocolateThe Family Block of Chocolate

Imagine if I gave you a family Imagine if I gave you a family block of chocolate which was made block of chocolate which was made up of 100 small squares every day.up of 100 small squares every day.

However as each day passed I However as each day passed I started to eat a few pieces before I started to eat a few pieces before I gave it to you. gave it to you.

On the second day I eat three On the second day I eat three pieces so you get 97 pieces.pieces so you get 97 pieces.

On the third day I eat six pieces, On the third day I eat six pieces, on the fourth I eat nine pieces!on the fourth I eat nine pieces!

So the amount of chocolate you So the amount of chocolate you get every day is 100+97+94+91+…get every day is 100+97+94+91+…

How much chocolate would you How much chocolate would you get altogether?get altogether?

Arithmetic SeriesArithmetic Series

This series of This series of numbers numbers (100+97+94+…) is (100+97+94+…) is called an arithmetic called an arithmetic series.series.

We can easily solve We can easily solve this problem with the this problem with the right Information right Information and tools!and tools!

Arithmetic Sequences and Arithmetic Sequences and seriesseries

Have a look at the following sequences:Have a look at the following sequences:

An Arithmetic series is a series of numbers in An Arithmetic series is a series of numbers in which each term is obtained from the which each term is obtained from the previous term by adding or subtracting a previous term by adding or subtracting a constant.constant.

The constant we add or subtract each time is The constant we add or subtract each time is called the common difference, “d” In our called the common difference, “d” In our chocolate example this was -3. The first term chocolate example this was -3. The first term is called “a” (here it was 100). is called “a” (here it was 100).

The letter “n” is used to denote the number The letter “n” is used to denote the number of termsof terms

AlgebraicallyAlgebraicallyT1

1st

term

T2

2nd termT3

3rd termT4

4th termTn

nth term

a a+ d a+2d a+3d ?

•So for any term, n

•Tn = a +(n-1) d

ExampleExample

Find the 20Find the 20thth term of the term of the sequencesequence

5,8,11,14,…5,8,11,14,…Here a = 5 and d= 3Here a = 5 and d= 3So TSo Tn n = 5+ (20-1)3= 5+ (20-1)3TTnn = 62 = 62

So what about our chocolate So what about our chocolate problem?problem?

We need to know how to We need to know how to sum an arithmetic sum an arithmetic sequence in order to sequence in order to solve our chocolate solve our chocolate problem. problem.

Here’s a neat proof to Here’s a neat proof to show you the formula.show you the formula.

The Sum of an Arithmetic The Sum of an Arithmetic Series- SSeries- Snn

Let the last term of an Arithmetic series be l.Let the last term of an Arithmetic series be l. SSnn= a + a+d + a+2d + a+3d + … + l-d + l Eqn = a + a+d + a+2d + a+3d + … + l-d + l Eqn

(1)(1) Now re-writing this backwards! Yes backwards!Now re-writing this backwards! Yes backwards!

SSnn = l + l-d + l-2d + l-3d + …. +a+d + a Eqn = l + l-d + l-2d + l-3d + …. +a+d + a Eqn (2)(2)

We are now going to add the two equations We are now going to add the two equations together- can you see why? What cancels out?together- can you see why? What cancels out?


So 2SSo 2Snn = lots of (a+l) but how = lots of (a+l) but how many if there are “n” terms? many if there are “n” terms?

Yes there are n lots of (a+l)Yes there are n lots of (a+l) This gives 2SThis gives 2Snn = n(a+l) = n(a+l)

But what if we don’t know the But what if we don’t know the last term?last term?

Sn = n/2 (a+l)


We can use l = a+ (n-1)d because We can use l = a+ (n-1)d because l is the nth term of the series so l is the nth term of the series so substitutingsubstituting

SSnn = n/2 ((a + a+(n-1)d)) = n/2 ((a + a+(n-1)d)) Which givesWhich gives SSnn = n/2 (2a + (n-1)d) = n/2 (2a + (n-1)d) Here’s a nice appletHere’s a nice applet

Example using the summation Example using the summation formulaformula

Find the sum of the first 22 Find the sum of the first 22 terms of the arithmetic series 6 terms of the arithmetic series 6 + 4+ 2 +….+ 4+ 2 +….

Using SUsing Snn = n/2 (2a + (n-1)d) = n/2 (2a + (n-1)d) SSnn = 11 ( 12 +21 (-2)) = 11 ( 12 +21 (-2)) SSnn = -330 = -330

How many pieces of How many pieces of chocolate?chocolate?

Our series for the choc0late problem Our series for the choc0late problem looks like this: 100+97+94+…..+looks like this: 100+97+94+…..+

Here there are 34 terms n=34 since Here there are 34 terms n=34 since 100/3100/3

So SSo S3434 = 34/2(200 +33(-3)) = 34/2(200 +33(-3)) SS3434 = 1717 = 1717 So you will eat 1717 pieces of chocolate So you will eat 1717 pieces of chocolate

after 34 days.after 34 days. How many pieces will you eat on the How many pieces will you eat on the

last day?last day?

A nice summary of AP’sA nice summary of AP’s

Here is a quick summary if you needHere is a quick summary if you need it. it.

Answers to sheet Answers to sheet