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IGCSE Sequences And Series

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Published in: Mathematics | Maths
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IGCSE Maths Sequences

Abdul W / Abu Dhabi

16 years of teaching experience

Qualification: PGCE Maths, B.Sc Mathematics

Teaches: Mathematics, Physics, Statistics, Maths, IGCSE/AS/AL, IB Exam Preparation

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  1. N/A
  2. 12 Here are the points that Carmelo scored in his last I I basketball games. 23 20 14 23 17 24 24 18 16 22 (a) Find the interquartile range of these points. Kobe also plays basketball. The median number of points Kobe has scored in his last I I games is 18.5. The interquartile range of Kobe's points is 10. (b) Which of Carmelo or Kobe is the more consistent points scorer? Give a reason for your answer. 21 (3) (1) (Total for Question 12 is 4 marks)
  3. 4 Here are the first five terms of a number sequence. 7 11 19 23 (a) Find an expression, in terms of n, for the nth tenu of this sequence. The nth term of a different number sequence is given by 80 (b) Write down the first 3 tenns of this sequence. (2) (2) Yuen says there are no numbers that are in both of the sequences. Yuen is collect. (c) Explain why (1) (Total for Question 4 is 5 marks)
  4. Introduction • This Chapter focuses on sequences and series • We will look at writing and using algebraic sequences • We will also be learning how to calculate the sum of a sequence
  5. Secacæ Senes .0: To understand arithmetic sequence and be able to use thearithmetic sequence formula
  6. The nth term The nth term of a sequence is sometimes known as the •general term'. You need to become familiar with the terminology of sequences in IGCSE/A-IeveI maths. Un = the nth term U 3 = the 3rd number in the sequence Sequences and Series U -56 Example 1 The nth term of a sequence is given by Un = 3n - 1. Work out the 1st, 3rd and 19th terms. 1st U =3n-1 UI -3-1 -2 U 19th Un -311-1 U -57-1 3rd n =3n-1 -8 6B
  7. Sequences and Series The nth term The nth term of a sequence is sometimes known as the •general term'. You need to become familiar with the terminology of sequences in A- level maths. Un = the nth term U 3 = the 3rd number in the sequence Example 3 The nth term of a sequence is given by: 2 n Work out the 20th term. 2 n 202 20+1 400 21 6B
  8. The nth term The nth term of a sequence is sometimes known as the •general term'. You need to become familiar with the terminology of sequences in A- level maths. Un = the nth term U 3 = the 3rd number in the sequence Sequences and Series 155=5n 31 = n Example 3 Find the value of n for which the formula Un =5n-2 has a value of 153. U n =5n-2 153=511-2 153 Add 2 Divide by 5 6B
  9. Sequences and Series The nth term The nth term of a sequence is sometimes known as the •general term'. You need to become familiar with the terminology of sequences in A- level maths. Un = the nth term U 3 = the 3rd number in the sequence Example 4 Find the value of n for which the formula Un=n2-7n+12 has a value of 72. =n2-7n+12 -n2-7n+12 72 — n2-7n-60 n = 12 or n = —5 But n has to be positive, so n : 12 un : 72 ubtract 72 Factorise 2 possible solutions 6B
  10. Sequences and Series The nth term The nth term of a sequence is sometimes known as the 'general term'. You need to become familiar with the terminology of sequences in A-level maths. 1) Form 2 equations using the information you have been given 2) Solve them simultaneously to find values for a and b Example 5 A sequence is generated by the formula Un = an+b Given that CJ3 5 and = 20, find the values of a and b. Un = an+b 20=8a+b 15 = 5a = a —4=b 20 1) 2) Un = an + b 20 = 8a -F b 6B
  11. Sequences and Series Arithmetic Sequences A sequence that increases by a constant amount is known as an arithmetic sequence. 3, 7, 11, 15, 19...(+4) 17, 14, 11, 8...(-3) a, a + d, a + 2d,a+ 3d...(+d) Example 1 Find the 10th, 50th and nth terms of the following arithmetic sequence. 3, 7, 11, 15, 19. First term: 3 Second term: 3 + 4 Third term: 3 +4 +4 Fourth term: 3 + 4 + 4 +4 a) 10th term 39 b) 50th term 3 + (49 x 4) : 199 C) nth term
  12. Sequences and Series Arithmetic Sequences A sequence that increases by a constant amount is known as an arithmetic sequence. 3, 7, 11, 15, 19...(+4) 17, 14, 11, 8...(-3) a, a + d, a + 2d,a+ 3d...(+d) Example 2 Find the number of terms in the following sequence. 7, 11, 15, . 143 -5 Increases in 4s 143-7: 136 34 -5 So there are 34 'jumps' 7, 11, 15, . 143 -5 There is always 1 more term than there are jumps -5 35 terms!
  13. 13 Sequences and Series 34 The nth term of an arithmetic sequence All arithmetic sequences take the form: Example 1 Find the 50th term of the following sequences: a) 4, 7, 10, 13. and b) 100, 93, 86, 79 a: 100 Ist term 2nd term 3rd term 4th term 5th term a) We can put this together as a relationship for the nth term of an arithmetic sequence... Where 'a' is the first term and 'd' is the common difference. a + (n —l)d -151 100 + x — (49 x —7) - -243
  14. The nth term of an arithmetic sequence All arithmetic sequences take the form: 13 Sequences and Series 34 + = 805 Example 2 For the following sequence, calculate the number of terms. 5+9+ 13 +17 21 + + 805 Ist term 2nd term 3rd term 4th term 5th term We can put this together as a relationship for the nth term of an arithmetic sequence... Where 'a' is the first term and 'd' is the common difference. -l)d -805 —1) x 4 = 805 —4 = 805 4n = 804 n =201 Substitute numbers in Work out the bracket Group together terms Subtract 1 Divide by 4 There are 201 terms in the sequence!
  15. 13 Sequences and Series 34 The nth term of an arithmetic sequence All arithmetic sequences take the form: Example 3 Given that the 3rd term of an arithmetic sequence is 20 and the 7th is 12: a) Work out the first term Ist term 2nd term 3rd term 4th term 5th term 3rd term a + (n —l)d = 20 _ 20 61+2d=20 7th term a + (n —l)d = 12 a + (7 —l)d = 12 a +6d We can put this together as a relationship for the nth term of an arithmetic sequence... Where 'a' is the first term and 'd' is the common difference. 1) 61+21=20 2) a+6d=12 2) - 1) 4d=-8 24 Substitute into 1) or 2)
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  17. Secacæ Senes .0: To be able to find the sum of n terms in the arithmeticsequenceusing the formula given Arithmetic series Sum to n terms S 2
  18. Arithmetic series terms. S • — The quadratic equation The + c • O a 0 are given by Trigonometry Of cone Curved Of Volume of cylinder = Curved surface area of cylinder = Area of trapezium In any triangle ABC Sine Rule S in Sin B Area or — C Volume Of prism mea of cross section length length Of sphere • — Surface of sphere
  19. Sequences and The Sum of an Arithmetic Series You need to be able to work out the sum of numbers in an arithmetic sequence. Add up the numbers from 1-100 13 Series 34 This method was discovered by Carl Friedrich Gauss (1777- 1855) while he was still in Primary School! Write out the same sequence backwards Add both sequences together We have 100 lots of 101 Halve that to get the actual total Arithmetic series s = 5050 Sum to n terms S n 2 — l)d]
  20. 13 Sequences and Series 34 The Sum of an Arithmetic Series As a general rule: (a+2d) + (a+3d) . 2Sn = 2Sn = = 2a + (n —l)d a + d + (a + (n —2)d) = 2a+d+11d-2d 261 -F (n -F 2a + (n — = 2a+nd2d" lots o Group the a's Multiply out the bracket Gedt)dth+cea + (n — Factorise E[2a + (n —l)d] 2 2 Divide by 2 If L is the last term in the series
  21. 13 34 Sequences and Series The Sum of an Arithmetic Series 2 2 a : the 1st term d : the common difference L : the last term Arithmetic series Sum to n terms S Example 1 Calculate the value of the first 100 odd numbers: n: 100 2 100 Substitute numbers in 2 Work out the inner bracket 50[2+198] sn = 50[200] 50 x 200 10,000
  22. 13 34 Sequences and Series The Sum of an Arithmetic Series 2 2 a : the 1st term d : the common difference L : the last term Arithmetic series Sum to n terms, S, Example 1 Calculate the value of the first 100 odd numbers: n: 100 100th term -5 a + (n — =199 sn- E[CI+L] 2 2 - 50[200] 10,000 Substitute numbers in
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  24. 13 34 Sequences and Series The Sum of an Arithmetic Series 2 2 a : the 1st term d : the common difference L : the last term Example 2 Find the number of terms needed for the sum of the following sequence to exceed 2000. 4+9+14+ 19. sn : 2000 a -4 d: 5 n 2 2000 = n [8 + 5n — 5] 4000 = n[3+5n] 4000= +311 5n2 +311-4000 Substitute numbers in Multiply by 2 Group together terms Multiply out the bracket Subtract 4000
  25. 13 34 Sequences and Series The Sum of an Arithmetic Series 2 2 a : the 1st term d : the common difference L : the last term Example 2 Find the number of terms needed for the sum of the following sequence to exceed 2000. 4+9+14+ 19. 5112 + 3n —4000 c : -4000 4ac 10 __3± 9 — (—80000) 10 10 Substitute numbers in Careful with negatives! n: 27.9 or -28.5 n : 28 ( need 28 terms to be over 2000)
  26. 23 The 4th term of an arithmetic series is 17. The 10th term of the same arithmetic series is 35. Find the sum of the first 50 terms of this arithmetic series. (Total for Question 23 is 5 marks)
  27. 23 The 4th term of an arithmetic series is 17. The 10th term of the same arithmetic series is 35. Find the sum of the first 50 terms of this arithmetic series. (Total for Question 23 is 5 marks) so
  28. 22 The 3rd tenn of an arithlnetic series, A, is 19 The sum of the first 10 tenns of A is 290 Find the 10th tenn of A.
  29. 22 The 3rd tenn of an arithlnetic series, A, is I The sum of the first 10 tenns of A is 290 Find the 10th tenn of A. 10
  30. 4. The first term of an arithmetic series is a and the common difference is d. The 18th term of the series is 25 and the 21st term of the series is 32— . 2 (a) Use this information to write down two equations for a and d. (2) (b) Show that a = —7.5 and find the value of d. (2) The sum of the first n terms of the series is 2750. —15n = 55 x 40. (c) (d) Show that n is given by Hence find the value of n. (4) (3) (Total 11 marks)
  31. 5. Sue is training for a marathon. Her training includes a run every Saturday starting with a run of 5 km on the first Saturday. Each Saturday she increases the length of her run from the previous Saturday by 2 km. (a) (b) (c) Show that on the 4th Saturday of training she runs I I km. (1) Find an expression, in terms of n, for the length of her training run on the nth Saturday. (2) Show that the total distance she runs on Saturdays in n weeks of training is n(n + 4) km. (3) On the n th Saturday Sue runs 43 km. (d) Find the value of n. (2) Find the total distance, in km, Sue runs on Saturdays in n weeks of training. (2) (Total 10 marks)
  32. 6. The first term of an arithmetic sequence is 30 and the common difference is —1.5 (a) Find the value of the 25th term. (2) The rth term of the sequence is O. (b) Find the value of r. (2) The sum of the first n terms of the sequence is Sn. (c) Find the largest positive value of Sn. (3) (Total 7 marks)
  33. 14. The first three terms of an arithmetic series are p, 5p — 8, and 3p + 8 respectively. (a) (b) (c) Show that p = 4. Find the value of the 40th term of this series. (2) (3) Prove that the sum of the first n terms of the series is a perfect square. (3) (Total 8 marks)
  34. 17. (a) (b) (c) An arithmetic series has first term a and common difference d. Prove that the sum of the first n terms of this series is — n[2a + (n — l)d]. 2 (4) The first three terms of an arithmetic series are k, 7.5 and k + 7 respectively. Find the value of k. (2) Find the sum of the first 31 terms of this series. (4) (Total 10 marks)
  35. The first term of an arithmetic series is (2t + l) where t > 0 The nth term of this arithmetic series is (14t — 5) The common difference of the series is 3 The sum of the first n terms of the series can be written as p(qt — l) r where p, q and r are integers. Find the value of p, the value of q and the value of r Show clear algebraic working.
  36. 2 June 2018 Paper 2H Question 23 The sum of the first 48 terms of an arithmetic series is 4 times the sum of the first 36 terms of the same series. Find the sum of the first 30 terms of this series. (Total for Question 2 is 5 marks)
  37. 34 Summary We have looked at sequences We have seen how to calculate a number in an arithmetic sequence We have also worked out the sum of a sequence We have also seen some of the notation which is used in sequences

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