Series

1. Series

2. Introduction • Chapter 5 of FP1 focuses on methods to calculate the sum of a series of numbers • The 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)

3. Teachings for Exercise 5A

4. Series You need to be able to understand and use the Σ notation This is the last value you put in to the formula (hence giving you the last 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 is the formula for the sequence you are calculating the sum of (the ‘nth’ term) This means ‘the sum of’ 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 first value you put in to the formula (hence giving you the first term to be summed) 5A

5. Series You need to be able to understand and use the Σ notation This is the last value you put in to the formula (hence giving you the last 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 is the formula for the sequence you are calculating the sum of (the ‘nth’ term) This means ‘the sum of’ 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 first value you put in to the formula (hence giving you the first term to be summed) In this example you are effectively calculating the sum of the terms from the 10th to the 20th 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… 5A

6. Series You 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: Write down the first 3 terms of, and the final term of, the sequence indicated by the notation below Put r = 1 in for the first term, r = 2 in for the second and so on… Put r = 3 in for the first term, r = 4 in for the second and so on… Stop after calculating the term for r = 10 Put ‘n’ in for the final term… 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 7 + 9 + 11 + ……………………………………+ (2n + 1) = 100 5A

7. Series You 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 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!) 5A

8. Series You 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) ) 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! 5A

9. Teachings for Exercise 5B

10. Series 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! 5B

11. Series 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! 5B

12. Series The sum of the numbers from 21 to 60… 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: 0 10 20 30 40 50 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… Notice the number here will always be one less than the one at the start! This is asking you to find the sum of the numbers from 21 to 60 Sum from 21 to 60 Sum from 1 to 60 Sum from 1 to 20 5B

13. Series 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: Sub in values for each part Calculate This is asking you to find the sum of the numbers from 21 to 60 So the sum of the numbers from 21 to 60 is 1620! 5B

14. Series 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) Remember the link between the starting number and the sum we subtract! The sum of the numbers from 1 to n The sum of the numbers we will be removing (k – 1) 5B

15. Series Write out the formula for the numbers you want… 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 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 Simplify 5B

16. Series • 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! 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. 5B

17. Teachings for Exercise 5C

18. Series 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! 5C

19. Series 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 5C

20. Series 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! 5C

21. Series 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! 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 Is given by this formula, where ‘n’ is the number of terms The sum of the first ‘n’ terms of this sequence Expand the bracket Group terms The two expressions are equivalent! Factorise 5C

22. Series Write as one sum subtract another 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 the formula separately for each sum Sub 50 into the first and 19 into the second Calculate each Finish off! 5C

23. Teachings for Exercise 5D

24. Series 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!) 5D

25. Series 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!) Calculate 5D

26. Series Write it as one sum subtract another 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 out the formula for the cubed sequence twice Sub in 40 for the first and 19 for the second Calculate Finish the sum! 5D

27. Series 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! 5D

28. Series 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! 5D

29. Series 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! 5D

30. Series 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! 5D

31. Teachings for Exercise 5E

32. Series 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! 5E

33. Series 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 Divide numerator and denominator by 2 ‘Clever factorisation’ Expand brackets Factorise! Group terms Take the factor 2 out of the bracket 5E

34. Series 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… 5E

35. Series 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! So we are finding the sum of the first 20 terms of the sequence! We can use the formula we were given! Sub in n = 20 Calculate 5E

36. Series 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: Write with the same denominator Combine Expand the bracket ‘Clever factorisation’ Expand the inner brackets Expand the bracket again Simplify (you should also factorise if possible) Write as 3 separate sums Write using the formulae above. Remember to include the coefficients! 5E

37. Series 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: Write as one sum subtract another Write the formulae out twice, one for each sum! Sub in 40 and 10 Calculate! 5E

38. Summary • We have seen how to calculate the sum of a series in various circumstances • We have practiced the correct series notation • We have also seen and used the ‘clever factorisation’ method for simplifying expressions!