Sequences and Series: Arithmetic and Geometric Sequences, Sums, and Sigma Notation, Exams of Calculus

Download Sequences and Series: Arithmetic and Geometric Sequences, Sums, and Sigma Notation and more Exams Calculus in PDF only on Docsity! Chapter 13 Sequences and Series 13 SEQUENCES AND SERIES Objectives After studying this chapter you should * be able to recognise geometric and arithmetic sequences; * understand Y notation for sums of series: * be familiar with the standard formulas for Dr, Lr? and Dr; 13.0 Introduction Suppose you go ona sponsored walk. In the first hour you walk 3 miles, in the second hour 2 miles and in each succeeding hour 3 of the distance the hour before. How far would you walk in 10 hours? How far would you go if you kept on like this for ever? This gives a sequence of numbers: 3, 2, 14,.. etc. This chapter is about how to tackle problems that involve sequences like this and gives further examples of where they might arise. It also examines sequences and series in general, quick methods of writing them down, and techniques for investigating their behaviour. Number of grains of corn shown Legend has it that the inventor of the game called chess was told to name his own reward. His reply was along these lines. ‘Imagine a chessboard. Suppose | grain of corn is placed on the first square, 2 grains on the second, 4 grains on the third, 8 grains on the fourth, and so on, doubling each time up to and including the 64th square. I would like as many grains of corn as the chessboard now carries.’ It took his patron a little time to appreciate the enormity of this request, but not as long as the inventor would have taken to use all the corn up. 245 Chapter 13 Sequences and Series Activity 1 (a) How many grains would there be on the 64th square? (b) How many would there be on the nth square? (c) Work out the numerical values of the first 10 terms of the sequence. 2°, 2°42! 2°42'42? etc. (d) How many grains are there on the chessboard? 13.1 Geometric sequences The series of numbers 1, 2, 4, 8, 16 ... is an example of a geometric sequence (sometimes called a geometric progression). Each term in the progression is found by multiplying the previous number by 2. Number of Money in years account (£) Such sequences occur in many situations; the multiplying factor 0 2000 does not have to be 2. For example, if you invested £2000 in an 1 2160 account with a fixed interest rate of 8% p.a. then the amounts of 2 2332.80 money in the account after | year, 2 years, 3 years etc. would be as 3 2159.42 shown in the table. The first number in the sequence is 2000 and 4 2720.98 each successive number is found by multiplying by 1.08 each time. Year Value (£) Accountants often work out the residual value of a piece of equipment by assuming a fixed depreciation rate. Suppose a piece ' a oo of equipment was originally worth £35 000 and depreciates in 2 28 350 value by 10% each year. Then the values at the beginning of each 3 25 515 succeeding year are as shown in the table opposite. Notice that 4 22 963.50 they too form a geometric progression. The chessboard problem in Activity 1 involved adding up PeBeP se ..+2, The sum of several terms of a sequence is called a series. Hence the sum 2°+2'+2°+4...+2% is called a geometric series (sometimes geometric progression, GP for short) Activity 2 Summing a GP In Activity 1 you might have found a formula for 1424274...42" 1. (a) Work out the values of 3°, 3° 431, 3°43 43° 246 Total growth in first nine weeks is 1.67(1.04° -1) $, = <17.67emto 4 sf. 104-1 Example: After how many complete years will a starting capital of £5000 first exceed £10 000 if it grows at 6% per annum? Solution After n years, the capital sum has grown to 5000 x (1.06)" When is this first greater than 10 000, n being a natural number? In other words, the smallest value of n is required so that 5000 x (1.06)" > 10000, ne N => (1.06)">2 Now take logs of both sides: nin1.06 > In2 In2 In1.06 > n>11.9 n> After 12 years, the investment has doubled in value. Activity 4 GP in disguise Chapter 13 Sequences and Series £5000. Ye £10, 000 (a) Why is this a geometric sequence? 1, -2, 4, -8, 16, ...? What is its common ratio? What is its nth term? What is S,? (b) Investigate in the same way, the sequence 1,-1,1,-1,.. How many years later? 249 Chapter 13 Sequences and Series Exercise 13A 1. Write down formulae for the nth term of these sequences: (a) 3, 6, 12, 24, ... (b) 36, 18, 9, 4.5, .. (c) 2, -6, 18, -54, ... (d) 90, -30, 10, -34, ... (e) 10, 100, 1000, ... (f) 6, -6, 6, -6, ... @mit,i,4 4° 12° 36° 108" 2. Use the formula for S, to calculate to 4 s.f. (a)5 + 10+ 20+... to 6 terms (b)4 + 12 + 36+... to 10 terms 1.1.1 (c) —-+—+—+...to 8 terms 3 6 12 (d) 100 - 20 + 4 —... to 20 terms (¢) 16 +17.6 + 19.36 +... to 50 terms (f) 26 — 16.25 + 10.15625 ... to 15 terms 13.2 Never ending sums Give the number (e.g. 12th term) of the earliest term for which (a) the sequence 1, 1.5, 2.25, ... exceeds 50; (b) the sequence 6, 8, 103, ... exceeds 250; 1 1 c) the sequence —,...g0es below —— ©) 4 10 g 1000 we 1 5 (a) For what value of n does the sum 50 + 60 + 72+... + 50x(1.2)"" first exceed 10002 (b) To how many terms can the following series be summed before it exceeds 2 000 000? 242.01 + 2.02005 +... Dave invests £500 in a building society account at the start of each year. The interest rate in the account is 7.2% p.a. Immediately after he invests his 12th instalment he calculates how much money the account should contain. Show this calculation as the sum of a GP and use the formula for S, to evaluate it. Many of the ideas used so far to illustrate geometric series have been to do with money. Here is one example that is not. If you drop a tennis ball, or any elastic object, onto a horizontal floor it will bounce back up part of the way. If left to its own devices it will continue to bounce, the height of the bounces decreasing each time. The ratio between the heights of consecutive bounces is constant, hence these heights follow a GP. The same thing is true of the times between bounces. Activity 5 Bouncing ball (a) A tennis ball is dropped from a height of 1 metre onto a concrete floor. After its first bounce, it rises to a height of 49cm. Call the height after the nth bounce h,. Find a formula for h, and say what happens to h, as n gets larger. (b) Under these circumstances the time between the first and second bounces is 0.6321 seconds. Call this r,. The next time, f,, is 0.71, and each successive time is 0.7 times the previous one. Find a formula for 1, 250 (c) If S,=1, +¢,, what does S, represent? What does S, mean? Calculate Sig, S4o and S59. How long after the first bounce does the ball stop bouncing altogether, to the nearest tenth of a second? Activity 5 gave an example of a convergent sequence. Convergence, in this context, means that the further along the sequence you go, the closer you get to a specific value. For example, in part (a) the sequence to the nearest 0.1 cm is 100, 49.0, 24.0, 11.8, 5.8, 2.8, 1.4, ... and the numbers get closer and closer to zero. Zero is said to be the limit of the sequence. Part (b) also gave a sequence that converged to zero. In part (c), the sequence of numbers S,, S,,53,... start as follows : 0.6321, 1.0746, 1.3844, 1.6012, 1.7530, ... You should have found that this sequence did approach a limit, but that this was not zero. Hence the series has a convergent sum, that is, the sum S), of the series also converges. The series 1, 2, 4, 8 ...... is a divergent sequence. It grows without limit as the number of terms increases. The same is true, in a slightly different sense, of the sequence 1, -2, 4, -8 cesses Any sequence that does not converge is said to be divergent. Activity 6 Convergent or divergent? For each of these sequences (i) write a formula for the n™ term; (ii) find whether the sequence converges; (iii) find whether the sum S, converges. 22 (a) 6,2, 3,3.... (b) 1, 1.5, 2.25, 3.375, ... () 4,-3.2, -2.. (d) 1, 1.01, 1.012, 1.0123, ... (e) 8,-9.6, 11.52, -13.824 ... Chapter 13 Sequences and Series 251 Chapter 13 Sequences and Series 13.3 Arithmetic sequences Geometric sequences involve a constant ratio between consecutive terms. Another important type of sequence involves a constant difference between consecutive terms; such a sequence is called an arithmetic sequence. My In an experiment to measure the descent of a trolley rolling down a slope a 'tickertape timer’ is used to measure the distance travelled in each second. The results are shown in the table. The sequence 3, 5, 7, 9, 11, 13 is an example of an arithmetic sequence. The sequence starts with 3 and thereafter each term is 2 more than the previous one. The difference of 2 is known as cm travelled the common difference. Second | in second It would be useful to find the total distance travelled in the first 6 seconds by adding the numbers together. A quick numerical trick for doing this is to imagine writing the numbers out twice, once forwards once backwards, as shown below 3 5 7 9 11 13 13, Il 9 7 5 3 Each pair of vertical numbers adds up to 16. So adding the two sequences, youhave 6x16 between them. Hence the sum of the original series is 1 7X (6x16) = 48. The sum of terms of an arithmetic sequence is called an arithmetic series or progression, often called AP for short. Activity 9 Distance travelled Use the example above of a trolley rolling down a slope to answer these questions. (a) Work out the distance travelled in the 20th second. (b) Calculate Sy the distance travelled in the first 20 seconds, using the above method. (c) What is the distance travelled in the nth second? (d) Show that the trolley travels a distance of n(n+2) cm in the first n seconds. 254 Example Consider the arithmetic sequence 8, 12, 16, 20... Find expressions (a) for u,,, (the nth term) (bo) for S.. In? Solution In this AP the first term is 8 and the common difference 4. (a) ete. u,, is obtained by adding on the common difference (n—1) times. > u, =8+4(n-1) =4n+4 (b) To find S,,follow the procedure explained previously: 8 12 haeeeeeeeees 4n 4n+4 Ant 4 An vieeeeeeee 12 8 Each pair adds up to 4n+12. There are n pairs. So 2S, =n(4n+12) =4n(n+3) giving S$, =2n(n +3). Exercise 13C Chapter 13 Sequences and Series 1. Use the ‘numerical trick’ to calculate 2. Find formulae for w, and S, in these sequences : (a)3 +74 114...427 (a) 1,4, 7, 10, ... (b)52 +46 +404... 44 (b) 12, 21, 30, 39, ... (c) the sum of all the numbers on a traditional (c) 60, 55, 50, 45, ... clock faces (d) 1 24. 4, She oe (4) the sum of all the odd numbers between 3. A model railway manufacturer makes pieces of Land 99. track of lengths 8 em, 10 cm, 12 cm, ete. up to and including 38 cm. An enthusiast buys 5 pieces of each length. What total length of track can be made? 255 Chapter 13 Sequences and Series The general arithmetic sequence is often denoted by a, at+d, a+2d, a+3d, ete. ... To sum the series of the first n terms of the sequence, S, =a+(a+d)+(a+2d)+ ... +(a+(n-1)d) Note that the order can be reversed to give S, =(a+(n-1)d)+(a+(n—2)d)+ ... +4 Adding the two expressions for S, gives 28, =(2a+(n-1)d)+(2a+(n-I)d)+ ... +(2a+(n-1)d) = n[2a +(n-1)d] So 5 (2a+(n-1)d) An alternative form for S, is given in terms of its first and last term, a and /, where l=a+(n-l)d since the nth term of the sequence is given by u, =a+(n-I)d. Thus 5, = 3(a+l) Example Sum the series 5 +9 + 13 +... to 20 terms. Solution This is an arithmetic sequence with first term 5 and common difference 4; so 20 Syy =F 2X5 +194) = 860 256 Chapter 13 Sequences and Series Solution ¥ (10-7)? =(10-1)? +(10-2)? + ... +(10-9)? r=l 2 2 2 =94+8 4+... 417 An alternative way of writing the same series is to think of it in reverse: 9 P42?4 0.4849 =y7 rl Example Express in Y notation 'the sum of all multiples of 5 between 1 and 100 inclusive’. Solution All multiples of 5 are of the form Sr,re IN. 100 =5 x20, so the top limit is 20. The lowest multiple of 5 to be included is 5X1. The sum is therefore 20 5+10+15+ ... +100= )5r rl Example Express in Y notation 'the sum of the first n positive integers ending in 3’. Solution Numbers ending in 3 have the form 10r+3, re N. The first number required is 3 itself, so the bottom limit must be r = 0. This means that the top limit must be n—1. Hence the answer is Fore3) (3413+ ... (102-7) r=0 [An alternative answer is ¥(10r-7)] rl 259 Chapter 13 Sequences and Series Exercise 13E 1. Write out the first three and last terms of: Is, 0 (a Yr (b) — ¥(2r-1) r=5 ral n 0 r-2 © br @ yA r= 100 ©) Pr-2" 1-6 v Shorten these expressions using Y notation. 11 1 (a) l4 z+ gt. t gy (b) 104+114+12+...4+50 (c) 1484274...40° (d) 143494+274...43"7 (e) 6+114+16+...4+(5n4 1) 13.5 More series Activity 11 (f) 144174 20+.,.4+62 (g) 5+50+5004...45x10". 1 2.3 20 (h) += +—+...4+ 6 12 20 21x22 Use ¥ notation to write: m of all natural numbers with two (b) the sum of the first 60 odd numbers; (c) the sum of all the square numbers from 100 to 400 inclusive; (a) the sum of all numbers between I and 100 inclusive that leave remainder 1 when divided by 7. Find alternative ways, using ) notation, of writing these: 19 (a) L(20-r) ral (a) Write down the values of (-1)°, (-1)', (-1)*, (-1)° ete. Generalise your answers. (b) Write down the first three terms and the last term of 10 ji) LX (-1) r=0 (i) Ecue{ r r=0 r+l (c) How can you write the series 1 a 100-100 x(0.8) +100 (0.8)? ... tonterms using } notation? Activity 12 Properties of ) (a) Calculate the numerical values of 5 5 5 5 rer XL (r+ r) ¥3r rs rs rl rel 1 260 Chapter 13 Sequences and Series (b) Ifu,,u,,...u,andv,, v,,..., v, are two sequences of numbers, is it true that n n n L(u,+v,)= Lup + Lv,? ral ral rsl Justify your answer. (c) Investigate the truth or falsehood of these statements: @ Lup», -(in.[£>,] rl ral ral rs n Gi) Yu, =| La, rl rel n n (iii) Yau, =a} Yu, | [ois any number.] rel rel Again, justify your answers fully. 5 What is the value of ¥(r+1)? r=l (d What is $12 and Er? ral ral Exercise 13F 1. Work out the numerical value of 3. If you know that 0 25 2 n wy wy) Ys Lu, =20and Ev, =64 ra rm 7 calculate where possible: © Pers @ Yoxas (a) Lu, +) to) La, ~ rm rl ral _ = >», ny a (e) Lon () Yong) (c) xu; (a) ray 2. Use Y notation to write these: (e) Lo, -u,) (f) E(u, —»,) rl rl 1 (a) l--—+---+4+ 2. --— 5 234 6 (g) Yu, (h) LD", rl rl (b) -1+4-9416- ... +144 (c) 12=-12x0.2+12%0.04- ... +12x(0.2)” 261