when n = 1, 2, 3, 4, 5, 6, This list represents the amount of dollars you have after n days. Note: The use of is read as and so on.

Transcription

1 Geometric eries Before we defie what is meat by a series, we eed to itroduce a related topic, that of sequeces. Formally, a sequece is a fuctio that computes a ordered list. uppose that o day 1, you have 1 dollar, ad every day you double your moey. The the fuctio f() = 2 geerates the sequece 1, 2, 4, 8, 16, 32,, whe = 1, 2, 3, 4, 5, 6, This list represets the amout of dollars you have after days. Note: The use of is read as ad so o. The idividual etries i a sequece are called the terms of the sequece. I our discussio, we are goig to assume that the terms i a particular sequece are real umbers. equeces ca be grouped ito two large classes based upo the umber of terms they iclude. A ifiite sequece is a fuctio that has the set of atural umbers as its domai. As the ame implies, it cotais a ifiite umber of terms. I the opeig example, the use of the without some umber o the ed implies that the sequece cotiues idefiitely, followig the prescribed patter. Of course, there is a iheret problem with assumig that moey ca be doubled forever. Istead, it makes sese to talk about doublig moey for a certai umber of days. ay, for = 1, 2, 3, 4, 5, 6, ad 7. I that case, the sequece geerated would be called a fiite sequece. Its domai is equal to a fiite set of atural umbers. (I this case, D = {1, 2,, 7}.) A commo otatio for sequeces is let a = f(). With this otatio, we say that a is the th term i the sequece. Example 1: Write out the first five terms a 1, a 2, a 3, a 4 ad a 5 of the followig sequeces. (a) a ( 1) (b) a si 2 (c) a 25 (d) 2(3) a 1 1

2 (a) a 1 = (-1) 1 = -1, a 2 = (-1) 2 = 1, a 3 = (-1) 3 = -1, a 4 = (-1) 4 = 1, a 5 = (-1) 5 = -1. (b) a1 si (1) 2 si 2 1, a2 si (2) 2 si 0 3 a3 si (3) 2 si 2 1, a4 2 5 a5 si (5) si , si (4) si 2 0, (c) a1 2(1) 5 3, a2 2(2) 5 1, a3 2(3) 5 1, a4 2(4) 5 3, a5 2(5) (d) a 1 2(3) 2(3) 2, a 2 2(3) 2(3) 6, a 3 2(3) 2(3) 18, a 4 2(3) 2(3) 54, a 5 2(3) 2(3) 162. It is worth otig that usig these formulas we would easily compute the 1,000 th term i the sequece. We would oly eed to plug i = ome sequeces are ot writte i terms of a explicit fuctio like those above. Istead, they may be defied recursively, ad hece are called a recursive sequece. That is, each term after the first few terms are defied i terms of what has come before it. If it happes that the terms i our sequece are multiplies of each other (as was the case i Example 1d), the we say that we have a geometric sequece. I Example 1d, the multiple betwee each term was 3. We call this umber the commo ratio ad it is usually deoted by a r. A geometric sequece ca be defied recursively based o the commo ratio betwee terms. That is, we have the relatioship a = ra 1. If we kow the startig term of our sequece, a 1, sice there is a commo ratio r betwee subsequet terms, we ca fid a explicit formula for the th term of the sequece. Let us work out a few terms ad try to discover the uderlyig patter. a 2 = ra 1 a 3 = ra 2 = r(ra 1 ) = r 2 a 1 a 4 = ra 3 = r(ra 2 ) = r(r(ra 1 )) = r 3 a 1 I geeral, we have 2

3 The th term of a Geometric equece I a geometric sequece with first term a 1 ad commo ratio r, the th term, a, is give by a = a 1 r 1 Example 2: Fid a formula for the geometric sequece give by 2, 1, 1/2, 1/4, 1/8, The first term is 2, so that is a 1. Notice that the commo ratio betwee subsequet terms is 1/2. o, we have that r = 1/2. Thus, a = 2(1/2) 1. Example 3: Fid a geeral term a for the followig geometric series if a 2 = 4 ad a 4 = 64. We kow that a 2 = a 1 r ad a 4 = a 1 r 3. o, if 4 = a 1 r ad 64 = a 1 r 3, we ca divide the two equatios to get 64/4 = (a 1 r 3 )/(a 1 r) = r 2, ad we see that r = 4. Pluggig that ito the first equatio, we 4 = a 1 (4), so a 1 = 1. Thus, a = 1(4) 1 = 4 1. Now that we have established what is meat by a sequece ad i particular a geometric series, we ca tur our attetio to a series. Recall, a sequece is a fuctio that computes a ordered list. A series, o the other had, is the summatio of elemets geerated by a sequece. Let us retur to our example with doublig moey that we opeed with. That is, suppose that o day 1, you have 1 dollar, ad every day you double your moey. Let s chage the sceario slightly. uppose o day 1 you have 1 dollar, but every day you are give twice the amout that you had the previous day. The fuctio that specifies how much moey you receive o the th days is give by f() = 2. This geerates the sequece 1, 2, 4, 8, 16, 32,, whe = 1, 2, 3, 4, 5, 6, uppose our iterest is how much moey you have after days. (Remember, the above values are oly how much you receive o a particular day. 3

4 You still get to keep your moey from the previous days!) We would eed to sum the values of our sequece up util day to aswer this questio. As was the case with sequece, series ca be grouped ito two large classes based upo the umber of terms they iclude. A ifiite series is the summatio of the terms i a ifiite sequece. A fiite series is the summatio of the terms i a fiite sequece. We shall cosider both types of series. We defie a geometric series as the summatio of the terms i a geometric sequece. We ca use the formula for the th term of the geometric sequece to develop a formula for the sum of the first terms i a geometric sequece. Recall, if a 1 was the first term i the geometric sequece with a commo ratio of r, the the formula for the th term i a geometric sequece is give by a = a 1 r 1. Let deote the sum of the first terms i a geometric sequece. The we have: = a 1 + a 1 r + a 1 r a 1 r 2 + a 1 r 1 Multiplyig this equatio by r, we have r = a 1 r + a 1 r a 1 r 1 + a 1 r ubtractig this equatio from, we have r = (a 1 + a 1 r + a 1 r a 1 r 2 + a 1 r 1 ) (a 1 r + a 1 r a 1 r 1 + a 1 r ) = a 1 a 1 r, sice all of the middle terms cacelled out. We ca factor a out of the terms o the left-had side ad a a 1 out of the terms o the right-had side to get (1 r) = a 1 (1 r ). Ad so, we have that um of the first terms of a Geometric equece If a geometric sequece has first term a 1 ad commo ratio r, the the sum of the first terms is give by 1 r a1 1 r, provided r 1 Notice that we eed to make the assumptio that r 1, sice we divided both sides by 1 r, would be 0 if r = 1. If it were the case that r = 1, the our geometric series actually reduces to a arithmetic series with d = 0. 4

5 Example 4: Fid the sum for the give values of (-2) 1 ; = 4, 7, ad 10. This is a geometric series with first term a 1 = 3 ad commo ratio r = ( 2) , 1 ( 2) ( 2) 3 129, 1 ( 2) ( 2) ( 2) Example 5: Fid the followig sum: This is a geometric series with the first term a 1 = 1 ad commo ratio r = 2. We are 8 1 (2) addig up the first 8 terms. Thus, we have o far, we have restricted our attetio to fiite series. There are some ifiite geometric series for which the sum is a fiite umber. The aciet Greek Zeo first proposed a variat of the followig problem. uppose a perso wats to walk through a forest that is oe mile wide. uppose he walks half the distace i a hour. The i the ext hour, walks half of the remaiig distace, ad cotiues i this maer. How far will the perso have walked? How log will it take the perso to leave the forest? The distace traveled by the perso is described by a ifiite series. Namely, Ituitio tells us that the perso will walk 1 mile (the total width of the forest). But sice the perso walks slower ad slower, it will take a ifiite amout of time to travel that distace. o, the perso ever actually leaves the forest! 5

6 Lookig at our formula for the fiite geometric series, otice that if r < 1, the as gets large, r approaches 0. That is, if r < 1, the lim r 0. Thus, we have 1r 10 a1 lim lim a1 a1 1 r 1r 1r. This is summarized as um of a ifiite Geometric equece The sum of the ifiite geometric sequece with first term a 1 ad commo ratio r is give by a1, provided r < 1. 1 r If r 1, the the sum either does ot exist or is ifiite. Example 6: how that the sum of the ifiite geometric sequece equals The first term i the sequece is a 1 = 1/2 ad the commo ratio is r = 1/2. Ad sice 1/2 < 1, we ca use the formula above to coclude that We ca use ifiite series to expressio fractios as summatios. This is doe by rewritig the fractio with a deomiator of 1 0.1, so our commo ratio will be 0.1, which correspods to oe decimal place. Example 7: Write 2/3 as a ifiite geometric series. Observe that This is i the form a ifiite geometric series with a 1 = ad r = 0.1. Thus, we have that