Cheat sheet on sequences and series, Cheat Sheet of Mathematics

Download Cheat sheet on sequences and series and more Cheat Sheet Mathematics in PDF only on Docsity! Sequences and Series Cheat Sheet Asequence is a list of terms. For example, 3, 6, 9, 12,15, ... Aseries is the sum of a list of terms. For example, 3 +6 +9+12 +15 + The terms of a sequence are separated by a comma, while witha series they are all added together. Definitions Here are some important definitions prefacing the content in this chapter: sequence is increasing if each term is greater than the previous. e.g. 4,9, 14, 19, ... Asequence is decreasing if each term is less than the previous. e.g. 5,4,3,2,1, Asequence is periodic if the terms repeat in a cycle; nsx = Un for some k, which is known as the order of the sequence. e.g. -3, 1, -3, 1, -3, ...is periodic with order 2. Arithmetic sequences An arithmetic sequence is one where there is a common difference between each term. Arithmetic sequences are of the form a, a+d, a+ 2d, a+3d, where a is the first term and d is the common difference. The n® term of an arithmetic series is given by: u, =a+(n—1)d Factorising out S,, from the LHS Arithmetic series and a from the RHS ‘An arithmetic series is the sum of the terms of an arithmetic sequence. The sum of the first m terms of an arithmetic series is given by S, = =[2a + (n—1)d] or Sn =F(a+D where a is the first term, d is the common difference and Lis the last term. You need to be able to prove this result. Here is the proof: Sxample 1: Prove that the sum ofthe fist n terms ofan arithmetic seriesis S, = #[2a-+ (n—1)d] We start by wsting the sum out normally [1], and then in reverse [2] [i] S,=0+ (etd) + (a42d) +4 (at (n— 2d) + (a+ (H—D)d) [2] 5, =(@+@-1d) + @+ n-2)d) +4 (@4 2d) HHA) 4a Adding [1] and [2] gives us (1+(2: 25, =n@a+(m-1a) n gl2a + (n—-1)d] Example 2: The fifth term of an arithmetic series is 33. The tenth term is 68. The sum of the first n terms is 2225. Show that 7n? + 3n — 4450 = 0, and hence find the value of n. Sth termis 33 -.a + 4d = 33 a (n° term formula) 10th term is 68.68 = a+ 9d p (n° term formula) Solving [1] and [2] simultaneously, we find that d= 7 and a = 5. Sum of first n terms is 2225 - —1)d] = 2225 Flto + @—1)(7)] = 2225 n(3 + 7n) = 2225(2) Tn? + 3n — 4450 =0 To find the value of n, we just need to solve the quadratic. Using the quadratic formula, we find that n= 25 or n= ~25.4, Since the term number must be a positive integer, we can conclude that m PMI eresources-tuition courses a HO, Geometric sequences The defining feature of a geometric sequence is that you must multiply by a common ratio, r, to get from one term to the next. Geometric sequences are of the form a, ar, ar’, ar”, ar’, where a is the first term in the sequence and r is the common ratio. The n* term of a geometric sequence is given by: Up, = ar”—1 Uk+a Uk+2 UK ~ Uke terms of the sequence are given in terms of an unknown constant. Part a of example 4 highlights this. It can help in many questions to use the fact that =. This is especially helpful when the Geometric series A geometric series is the sum of the terms of a geometric sequence. The sum of the first n terms of a geometric series is given by: a(i—r") 5, =——— 1-r by multiplying the top and bottom of the fraction by -1, we can also use _ag"=1) S, nad You need to be able to prove this result. Here is the proof: Example 3: Prove that the sum of the first n terms of a geometric series is S, Sys atar tar? $e¢ar" multiplying the sur by r 15, ar $ar? + ar? +4 ar" Subtracting [2] from [1] Sn —1Sn = a— ar" = S,(1-1) =a(1—r") Factoring out Sy, and a Dividing by 1 —r Since division by zero is undefined, this formula is invalid when r Sum to infinity The sumto infinity of a geometric sequence is the sum of the first n terms as n approaches infinity. This does not exist for all geometric sequences. Let’s look at two examples: 244484 164 32+- Each term is twice the previous (i.e. r = 2). The sum of such a seriesis not finite, since each term is bigger than known as a divergent sequence. oaratyhyl +ltgtgtst Here, each term is half the previous (ie. r = 3). The sum of such a series terms will tend to 0. This is known as a convergent sequence. ce as n becomes large, the Ageometric sequence is convergent if and only if |r| <1. The sum to infinity of a geometric sequence only exists for convergent sequences, and is given by: © wwwanteducation OO PMTEducation Pure Year 2 Example 4; The first three terms of a geometric series are (k-6), k, (2k45), where k is @ positive constant. 2) Show that k® — 7k — 30 = 0 b) Hence find the value ofk. €)Find the common ratio of this series and hence calculate the sum of the first 10 terms. d) Find the sum to infinity for a series with first term k and common ratio ~ 2) Using the fact that 5 = BE = kts ie tegn = (2k + 5)(k —6) 2k? — 7k —30 = k?— 7k — 30 = 0s required. (k-10(k+3)=0 > k=10, k Since we are told k is positive, we can conclude k = 10. 20219) _ 60°85) _ 25499,6 25400 103 s.f ‘Cross-multipiying and simplifying: b) Solving the quadratic: ¢) From part a : Sy dja =10andr Recurrence relations Arecurrence relation is simply another way of defining a sequence. With recurrence relations, each term is given as a function of the previous. For example, t41 = Un +4, ty = 1 represents an arithmetic sequence with first term 1 and common difference 4, Inorder to generate a recurrence relation, you need to know the first term. Example 5: The sequence with recurrence relation uy41 = pux +4, Us = 5, where p isa constant andq = 13, is periodic with order 2, Find the value of p. We know the order is 2. Sofi, = 5,then u, uy = py +13 = Sp +13, too. Finding ws: Ug = pu, + 13 = p(Sp + 13) + 13 Equating to 5: p(Sp +13) +13 = 5 simplifying Sp? + 13p + 8=0 Solving the quadratic by factoring Gp +8) +D We get 2 values, one of whichis correct, p=-lorp=-16 Substitute p = —1.6and p = —1 into the Substituting p = —1.6 into the recurrence relation gives a recurrence relation separately to see which one sequence where each term is 5 and so does not have order 2. correctly corresponds to aperiodic sequenceof Using p = —1 does give us a periodic sequence with order 2 order 2 however, sop = —1. Sigma notation You need to be comfortable solving problems where series are given in sigma notation. Below is an annotated example explai how the sigma notation is used. ‘This tellsus the last value of efor our sequence, ie. our last term willbe 7 2(20) = ~33, (20) 20 Di R28 434 E arithmetic series with fst term 5 Inputting r = 1 into this expression gives us the first term, r= 2gives us the second term, and so forth Tek beak orwtere sur —_ seriesstarts,.e.our fist term s 7-20) =5 and cammon ference -2 Ifyou are ever troubled by a series given in sigma notation, it is a goad idea to write out the first few terms and analyse the series that way. Modelling with series Geometric and arithmetic sequences are often used to model real-life scenarios. Consider the amount of money in a savings account; this can be modelled by a geometric sequence where r represents the interest paid at the end of each year and ais the amount of money in the account at the time of opening. You need to be able to apply your knowledge of sequences and series to questions involving real-life scenarios. Itis important to properly understand the context given to you, so take some time to read through the question more than once. Example 6: A virus is spreading such that the number of people infected increases by 4% each day. Initially 100 people were diagnosed with the virus. How many days will it be before 1000 are infected? = 100 andr = 1.04, We are really just trying to find the smallest value of n such that U,, > 1000. 1. divide both side by 100 U,, = 1900 < 100(1.04") 10 < (1.04") 2. take logs of both sides log(10) < a log (1.04) 3. divide both side by og(1.08) to solve for n (note that log(10) = 2) oO 4, round your answer up OS ECT