Series - PowerPoint Presentation

Slide1

SeriesSlide2

IntroductionChapter 5 of FP1 focuses on methods to calculate the sum of a series of numbersThe main focus is based around summing the first ‘n’ natural numbers of a given power

You will also become familiar with the proper series notation (you may already have seen this if you have covered Arithmetic and Geometric sequences from C1 and C2)Slide3

Teachings for

Exercise 5ASlide4

SeriesYou need to be able to understand and use the Σ notation

5A

This means ‘the sum of’

This is the first value you put in to the formula (hence giving you the first term to be summed)

This is the last value you put in to the formula (hence giving you the last term to be summed)

This is the formula for the sequence you are calculating the sum of (the ‘nth’ term)

This means ‘the sum of’

This is the first value you put in to the formula (hence giving you the first term to be summed)

This is the last value you put in to the formula (hence giving you the last term to be summed)

This is the formula for the sequence you are calculating the sum of (the ‘nth’ term)Slide5

SeriesYou need to be able to understand and use the Σ notation

5A

This means ‘the sum of’

This is the first value you put in to the formula (hence giving you the first term to be summed)

This is the last value you put in to the formula (hence giving you the last term to be summed)

This is the formula for the sequence you are calculating the sum of (the ‘nth’ term)

This means ‘the sum of’

This is the first value you put in to the formula (hence giving you the first term to be summed)

This is the last value you put in to the formula (hence giving you the last term to be summed)

This is the formula for the sequence you are calculating the sum of (the ‘nth’ term)

In this example you are not told how many terms there are, so the answer will be in terms of ‘n’, the number of terms…

In this example you are effectively calculating the sum of the terms from the 10

th

to the 20

th

Slide6

SeriesYou need to be able to understand and use the Σ notation

Write out the terms defined by the following notation, and hence calculate the sum of the series:

5A

Put r = 1 in for the first term, r = 2 in for the second and so on…

Stop after calculating the term for r = 10

1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19

= 100

Write down the first 3 terms of, and the final term of, the sequence indicated by the notation below

Put r = 3 in for the first term, r = 4 in for the second and so on…

Put ‘n’ in for the final term…

7

+ 9 + 11 + ……………………………………+ (2n + 1)Slide7

SeriesYou need to be able to understand and use the Σ notation

Write the sequence below using the

∑ notation.

You need the formula for the sequence and the values to put in for the first and final terms!

Formula for the sequence = ‘3n – 1’ (from GCSE

maths

!

Substituting 1 in will get 2 – the first term

Substituting 8 in will get 23 – the final term

5A

2 + 5 + 8 + 11 + 14 + 17 + 20 + 23

It is usually possible to do this several ways

 The notation below will give the same sequence (although is not as easy to work out!)Slide8

SeriesYou need to be able to understand and use the Σ notation

Write the sequence below using the

∑ notation.

You need the formula for the sequence and the values to put in for the first and final terms!

Formula for the sequence = n(n+1) (multiply the term by the number one bigger than it!

Substituting 1 in will get the first term

To get the final term, we sub in (n – 2)

5A

1 x 2 + 2 x 3 + 3 x 4 + 4 x 5 + …… + (n – 1)(n – 2)

)

Replace n with ‘n – 2’

Simplify

Rewrite

 This is the final term!Slide9

Teachings for

Exercise 5BSlide10

Series5B

You need to be able to use the formula for the sum of the first n natural numbers

The sum of the first ‘n’ natural numbers:

1 + 2 + 3 + 4 + …… + n

is an arithmetic sequence with ‘n’ terms, with a = 1 and l (last term) = n

The formula for the sum of the first ‘n’ natural numbers is shown to the right:

This is just saying the sum of the first ‘n’ natural numbers

 The formula for the sequence is just ‘r’ as when you substitute in the term number, that is the term itself!

This is the

formula

for the sum of the first ‘n’ natural numbers



For example if you wanted the sum of the first 30 numbers, let n = 30 and calculate the answer!

You DO NOT get given this formula on the exam!Slide11

Series5B

You need to be able to use the formula for the sum of the first n natural numbers

Calculate the sum of the series indicated below:

We want the sum of the first 50 terms, so n = 50

Calculate

It is fine (and sometimes easier) to use the formula in this form!Slide12

Series5B

You need to be able to use the formula for the sum of the first n natural numbers

Calculate the sum of the series indicated below:

This is asking you to find the sum of the numbers from 21 to 60

0

10

20

30

40

50

60

The sum of the numbers from 21 to 60…

… Will be equal to the sum of the numbers from 1 to 60, subtract the numbers from 1 to 20…

-

The notation will look like this…

Sum from 21 to 60

Sum from 1 to 60

Sum from 1 to 20

Notice the number here will always be one less than the one at the start!Slide13

Series5B

You need to be able to use the formula for the sum of the first n natural numbers

Calculate the sum of the series indicated below:

This is asking you to find the sum of the numbers from 21 to 60

Sub in values for each part

Calculate

So the sum of the numbers from 21 to 60 is 1620!Slide14

Series5B

You need to be able to use the formula for the sum of the first n natural numbers

This is the general form for the problem you have seen (where you sum the numbers of a section of natural numbers, not starting on 1)

The sum of the numbers from 1 to n

The sum of the numbers we will be removing (k – 1)

Remember the link between the starting number and the sum we subtract!Slide15

Series5B

You need to be able to use the formula for the sum of the first n natural numbers

Show that:

This type of question can look confusing but in reality you proceed as before

We want the sum of the natural numbers from 5 to 2N – 1

This will be the sum of the natural numbers from 1 to 2N – 1, subtract the numbers from 1 to 4.

Write out the formula for the numbers you want…

Write out the formula twice as you will need it for both!

Sub (2N – 1) in for the 1st, and 4 in for the 2nd

Simplify or calculate where possible

Expand brackets and group up

SimplifySlide16

Series5B

You need to be able to use the formula for the sum of the first n natural numbers

Show that:

This type of question can look confusing but in reality you proceed as before

We want the sum of the natural numbers from 5 to 2N – 1

This will be the sum of the natural numbers from 1 to 2N – 1, subtract the numbers from 1 to 4.

But why N ≥ 3?

Remember that you are told k = 5, meaning the first number in the sequence we are summing should be 5!

If you use the value of N = 3, the sum of the sequence is 5, hence 3 is the lowest number we can put in!Slide17

Teachings for

Exercise 5CSlide18

Series5C

You need to be able to split up parts of a sequence and sum them separately

You can split up series sums of the form:

into 2 separate ‘series sums’ as follows:

This allows you to then use the sum formulae for the sequence overall!

 Slide19

Series5C

You need to be able to split up parts of a sequence and sum them separately

Show that:

Can be written as:

If we wrote out the first few terms of this sequence…

This is equal to the sum of the multiplied terms, added to the sum of the 2s

We can ‘

factorise

’ the 3 out of the multiplied terms and

factorise

a 2 from the added terms…

This is 3 multiplied by the sum of the first ‘n’ numbers represented by the formula ‘r’

This is 2 multiplied by ‘n’ 1s

 Slide20

Series5C

You need to be able to split up parts of a sequence and sum them separately

Evaluate:

You need to split this up and sum the parts separately!

Split into two separate parts as you have seen

Write the formulae for the sums. Remember the 3 at the start of the first one!

 We will also have ‘n’ lots of 1

Sub in n = 25 (25 terms to add up)

Calculate

So the first 25 terms of the sequence with the formula (3r + 1) will add up to 1000!

 Slide21

Series5C

You need to be able to split up parts of a sequence and sum them separately

Show that:

In this case you should proceed as normal, but use ‘n’ instead!

The sum of the first ‘n’ terms of this sequence

Is given by this formula, where ‘n’ is the number of terms

Split up as two separate sums

Remember the 7 on the first expression!

 We also have n lots of 4

Write ‘4n’ as fraction over 2 (for grouping)

Group terms

Expand the bracket

Group terms

Factorise

The two expressions are equivalent!Slide22

Series5C

You need to be able to split up parts of a sequence and sum them separately

Show that:

Hence, calculate the value of:

Here, you can use the formula you’re given – remember that this will be the sum of the first 50 terms subtract the sum of the first 19!

Write as one sum subtract another

Write the formula separately for each sum

Sub 50 into the first and 19 into the second

Calculate each

Finish off!Slide23

Teachings for

Exercise 5DSlide24

Series5D

You need to be able to calculate the sum of a sequence based on powers of 2 and 3

The sum of a sequence of squared numbers is given as follows:

And the formula for the sum of a sequence of cubes is:

You will see where these come from in chapter 6!

You get given these formulae in these forms in the exam booklet!

(Remember you do not get the formula for a linear sequence!)Slide25

Series5D

You need to be able to calculate the sum of a sequence based on powers of 2 and 3

Evaluate:

Write out the formula for a squared sequence

Sub in n = 30 as we want 30 terms

Simplify the numerator (if necessary!)

CalculateSlide26

Series5D

You need to be able to calculate the sum of a sequence based on powers of 2 and 3

Evaluate:

Remember for this one you need the sum of the first 40 terms, subtract the first 19 terms!

Write it as one sum subtract another

Write out the formula for the cubed sequence twice

Sub in 40 for the first and 19 for the second

Calculate

Finish the sum!Slide27

Series5D

You need to be able to calculate the sum of a sequence based on powers of 2 and 3

Find:

This one is more algebraic but you still approach it the same way!

The first value we put in the sequence will be ‘n + 1’

The final value we put in will be ‘2n’

So we want the sum of the first ‘2n’ terms, subtract the first ‘n’ terms (same as if we were using numbers!)

Write out the formula twice

Sub ‘2n’ into the first and ‘n’ into the second

You can write this as one fraction

This is the key step – you can

factorise

as n(2n+1) is common to both terms!

Expand the terms in the square bracket

Simplify the square bracket (which can now be written as a ‘normal’ bracket!)

The

factorising

step is crucial here – otherwise you will end up trying to

factorise

a cubic which can take a long time!Slide28

Series5D

You need to be able to calculate the sum of a sequence based on powers of 2 and 3

Find:

Verify that the result is correct for n = 1, 2 and 3

(This can show the formula is working, although in reality isn’t a proof in itself!)

If n = 1

The first number we put in is 2, which is also the last number we put in

Sequence



So the numbers in the sequence just add up to 4!



Let’s check the formula!Slide29

Series5D

You need to be able to calculate the sum of a sequence based on powers of 2 and 3

Find:

Verify that the result is correct for n = 1, 2 and 3

(This can show the formula is working, although in reality isn’t a proof in itself!)

If n = 2

The first number we put in is 3, and the last number we put in in 4

Sequence



So the numbers in the sequence add up to 25



Let’s check the formula!Slide30

Series5D

You need to be able to calculate the sum of a sequence based on powers of 2 and 3

Find:

Verify that the result is correct for n = 1, 2 and 3

(This can show the formula is working, although in reality isn’t a proof in itself!)

If n = 3

The first number we put in is 4, and the last number we put in in 6

Sequence



So the numbers in the sequence add up to 77



Let’s check the formula!

So the formula seems to be working fine!Slide31

Teachings for

Exercise 5ESlide32

Series5E

You need to be able to use all you have learnt to calculate the sum of a more complex series, made up of several terms

As you saw in section 5C, you can take out a coefficient of a term in order to sum it.

You can also do this with the sums for r2

and r3.

For example:

 You need to remember to include the coefficient in the formula though!

 Slide33

Series

5E

You need to be able to use all you have learnt to calculate the sum of a more complex series, made up of several terms

Show that:

Write as separate sums

Write the formula for each part in terms of n

Write all with a common denominator

Group up

‘Clever

factorisation

’

Expand brackets

Group terms

Take the factor 2 out of the bracket

Divide numerator and denominator by 2

Factorise

!Slide34

Series5E

You need to be able to use all you have learnt to calculate the sum of a more complex series, made up of several terms

Show that:

Hence, calculate the sum of the series:

4 + 10 + 18 + 28 + 40 … … … + 418

You can see that this formula gives us the sequence we are trying to find the sum of!

(The 0 at the start will not affect the sum so can be ignored!)

We need to know how many terms there are, so have to find the value for r which gives a term with a value of 418…Slide35

Series5E

You need to be able to use all you have learnt to calculate the sum of a more complex series, made up of several terms

Show that:

Hence, calculate the sum of the series:

4 + 10 + 18 + 28 + 40 … … … + 418

Subtract 418

Factorise

2 answers, only 1 is possible though!

We can use the formula we were given!

So we are finding the sum of the first 20 terms of the sequence!

Sub in n = 20

CalculateSlide36

Series5E

You need to be able to use all you have learnt to calculate the sum of a more complex series, made up of several terms

Find a formula for the sum of the series:

Expand the bracket again

Expand the bracket

Write as 3 separate sums

Write using the formulae above. Remember to include the coefficients!

Write with the same denominator

Combine

‘Clever

factorisation

’

Expand the inner brackets

Simplify (you should also

factorise

if possible)Slide37

Series5E

You need to be able to use all you have learnt to calculate the sum of a more complex series, made up of several terms

Find a formula for the sum of the series:

Hence, calculate the following:

Sub in 40 and 10

Calculate!

Write as one sum subtract another

Write the formulae out twice, one for each sum!Slide38

SummaryWe have seen how to calculate the sum of a series in various circumstancesWe have practiced the correct series notation

We have also seen and used the ‘clever factorisation’ method for simplifying expressions!