THE SUM OF A GEOMETRIC SERIES

Transcription

1 THE SUM OF GEOMETRIC SERIES S T T T... T T a ar ar ar ar (1) Multiplyig lie (1) by r gives 1 rs ar ar... ar ar ar () () (1) rs S ar a Makig S the subject: S ( r1) a( r 1) S ar ( 1) r 1 The above form of S is most commoly used where r > 1. Or, multiplyig top ad bottom a(1 r ) by 1 gives S which is most commoly used where r < 1. 1 r Note: The above guide does ot eed to be adhered to. Either formulae is suitable i each case, as they are equivalet. S S ar ( 1), r 1 r 1 a(1 r ), r 1 1 r The School For Excellece 011 Trial Exam Preparatio Lectures Uit Maths Book 3 Page 1

2 QUESTION 16 Fid the sum of the geometric series Solutio 1 a log 3 1 log 9 log 3 r 1 1 log log log log log to 8 terms ad 8 S ar ( 1) r 1 The School For Excellece 011 Trial Exam Preparatio Lectures Uit Maths Book 3 Page

3 RITHMETIC ND GEOMETRIC MENS If umber of arithmetic meas (let s call them m,,..., 1 m m ) are iserted betwee two existig umbers x ad y, the resultig sequece of x, m1, m,..., m, y is a.p., cotaiig terms. Similarly, if umber of geometric meas are iserted betwee x ad y, a G.P. is formed, agai with terms. QUESTION 17 Isert 5 arithmetic meas betwee 15 ad -1. Solutio The resultig.p. has 7 terms 5, where T1 15 ad T7 1. i.e. a 15 (1) ad a6d 1 () T7 a 71 d a 6d.) (Note that Substitute (1) ito (): 15 6d 1 6d 36 d 6 T 1 15 T 1569, T 96 3 ad so o. 3 Hece the five arithmetic meas are 9,3, 3, 9, 15. The School For Excellece 011 Trial Exam Preparatio Lectures Uit Maths Book 3 Page 3

4 QUESTION 18 Isert 4 geometric meas betwee Solutio The resultig G.P. has 6 terms i.e. 6 a (1) 4 6 ad , where T 1 ad T ad ar () (Note that T ar ar ). The School For Excellece 011 Trial Exam Preparatio Lectures Uit Maths Book 3 Page 4

5 LIMITING SUM y geometric series with a commo ratio r such that 1 < r < 1 i.e. r 1, has a limitig sum. This meas that there is some fiite sum (called S ) that caot be exceeded, eve if the series is added forever. S a 1 r oly where 1 < r < 1 Proof: S a(1 r ) lim 1 r But if 1r 1 the as, r 0 Hece S a(1 0) a 1 r 1 r lterative Proof: Let S a ar ar ar 3... i.e. S a r a ar ar... Note that although this series has oe less term tha the previous lie, oe less tha ifiity is still ifiity! i.e. S The School For Excellece 011 Trial Exam Preparatio Lectures Uit Maths Book 3 Page 5

6 QUESTION 19 youg tree of height 10 cm grows 30 cm i its first year after platig, 0 cm i its secod year, ad two thirds of its previous year s growth i each subsequet year. Fid the maximum height of the tree. Solutio 1 The limitig height is the ifiite sum of (cm). 3 Note however that the 10 does ot belog to the remaiig patter, which is a geometric series with a 30 ad r. 3 1 Limitig height S cm QUESTION 0 Write as a ifiite series of fractios, ad hece express i ratioal form. (Note: ratioal form is p q where p ad q are itegers with o commo factor other tha 1.) Solutio The School For Excellece 011 Trial Exam Preparatio Lectures Uit Maths Book 3 Page 6

7 FINNCIL PPLICTIONS COMPOUND INTEREST Imagie that $300 was to be icreased by 5 % i iterest each year for 7 years. I all but the first year, we would gai iterest ot just o our origial ivestmet of $300, but also o ay iterest eared i the meatime. This is called compoud iterest (as opposed to simple iterest where the iterest is calculated oly o the origial ivestmet, for ay year). The most coveiet way of icreasig a amout by 5 % is to multiply by Why? Because % 105% 1.05 We eed to do this each year for 7 years. i.e. $300 x 1.05 x 1.05 x x years $300 x This ca be expressed i the geeral case by: Where amout of fial balace P pricipal (origial) ivestmet r iterest rate as a percet umber of calculatios r P(1 ) 100 If the iterest was calculated every 6 moths (half yearly) istead, the r becomes 5%.5% ad the umber of calculatios is 7 14, i.e. $300 x The two aswers are the $4.13 ad $43.89 respectively (i.e. for yearly compoudig ad the for six mothly compoudig). Note the differece. WTCH OUT! Make sure that the iterest rate you use is cosistet with the frequecy of calculatios e.g 1% p.a. should appear as 1% if compouded yearly, but as 6% if compouded half yearly, 3% if compouded quarterly ad so o. The School For Excellece 011 Trial Exam Preparatio Lectures Uit Maths Book 3 Page 7

8 SUPERNNUTION Superauatio questios, i geeral, are cocered with a situatio where a perso ivests a set amout each time period (ormally deposited at the begiig of each year), with iterest beig compouded, ad the accumulated total the beig withdraw at the ed of the required legth of time. We ca track the progress of each idividual deposit, as it accumulates iterest, idepedet of the other deposits, almost as if each deposit was assiged its ow accout. QUESTION 1 $1000 was to be deposited at the begiig of each year, for 10 years, earig iterest at the rate of 7 % per aum. Fid the total value of these deposits at the ed of the 10 years. Solutio % 107% 1.07 ddig these idividual amouts right to left (for coveiece): Let be the value of the th ivestmet (1.07) ( ) ( ) ( ) [ ] This is a G.P. with a1.07, r 1.07, 10 The School For Excellece 011 Trial Exam Preparatio Lectures Uit Maths Book 3 Page 8

9 $14, Note also that the iterest eared is: QUESTION Cosider a similar situatio to the above, except after 3 years the iterest rate is icreased from 7 % p.a. to 8 % p.a. Fid ow the total value of the ivestmets. Solutio The School For Excellece 011 Trial Exam Preparatio Lectures Uit Maths Book 3 Page 9

10 TIME PYMENTS There are may variatios possible i time paymets ot just i the umbers of course, but i the frequecy of paymets, frequecy of iterest calculatios ad so o. Cosider these examples. QUESTION 3 loa for $30,000 is to be paid off with equal istalmets of $400, paid at the ed of each moth. Iterest is calculated at a rate of 1 % p.a. (i.e. 1 % per moth), based o the reducig balace, ad debited at the begiig of each moth. Fid the balace owig after 1,, 3 moths. Solutio The School For Excellece 011 Trial Exam Preparatio Lectures Uit Maths Book 3 Page 30

11 QUESTION 4 Now imagie similar circumstaces to the above, except where the mothly repaymets are $R, but where we wish to fully repay the loa i 10 years. i.e. 10 moths. How do we fid R? Solutio % 101% R R1.01 R (Note the reversal i order of the last two terms this gives a icreasig series which I thik looks more maageable) R R R (Note here the use of the secod last lie of, ot the last lie, for the balace carried forward to the start of the third moth). (gai, I reversed the order.) It is safe to predict that 10 will be: R Note that although the last term s power is 119, there are still terms i the bracket, 0 sice ad coutig from meas 10. Furthermore, each term i the bracket correspods to a, ad 10 moths meas 10 paymets. fter 10 moths, the balace is to be 0. That is, the loa is to be fully repaid R R This is a geometric series with a1, r 1.01, 10 The School For Excellece 011 Trial Exam Preparatio Lectures Uit Maths Book 3 Page 31

12 10 1(1.01 1) R R (1.01 1) R $ ( earest cet) 10 Note the that the total iterest charged o this loa would be: I $ $30000 $ Because we are payig the same istalmet each moth, yet the iterest each moth is calculated o the reducig balace, less of the istalmet for successive moths is used to offset the iterest, ad more therefore is used to reduce the pricipal. Hece, if we were to graph the balace agaist the umber of moths, the graph would look like: Now, imagie a similar situatio to the previous example, but where the first six moths are iterest free. The: 0 $30, The School For Excellece 011 Trial Exam Preparatio Lectures Uit Maths Book 3 Page 3

13 R 1.01 R R 1.01 R R R1.01R 1.01 R R R R R R1.01 R1.01R 1.01R R R R R R It is safe to predict that 10 will be: R gai, let 10 0 : R R This is a G.P. with a1, r 1.01, 114 1(1.01 1) R R Note that dividig by 0.01 is equivalet to multiplyig by 100. R $406.30earest cet The School For Excellece 011 Trial Exam Preparatio Lectures Uit Maths Book 3 Page 33

14 FURTHER QUESTIONS QUESTION 5 (HSC 199) (i) For what value of r does the geometric series a ar ar... have a limitig sum? For these values of r write dow the limitig sum. (ii) Fid a geometric series with commo ratio 1 w that has a limitig sum 1 1 w. The School For Excellece 011 Trial Exam Preparatio Lectures Uit Maths Book 3 Page 34

15 QUESTION 6 (HSC 199) timber worker is stackig logs. The logs are stacked i layers, where each layer cotais oe log less tha the layer below. There are five logs i the top layer, six logs i the ext layer, ad so o. There are layers altogether. (i) Write dow the umber of logs i the bottom layer. (ii) Show that there are 1 ( 9) logs i the stack. Solutio The School For Excellece 011 Trial Exam Preparatio Lectures Uit Maths Book 3 Page 35

16 QUESTION 7 (HSC 004) Cosider the geometric series 4 1 ta ta... (i) (ii) Whe the limitig sum exists, fid its value i simplest form. For what value of i the iterval does the limitig sum of the series exist? Solutio The School For Excellece 011 Trial Exam Preparatio Lectures Uit Maths Book 3 Page 36

17 QUESTION 8 (HSC 1993) Tap m First Trough 3m Secod Trough 3m Third Trough tap ad water troughs are i a straight lie. The tap is first i lie, metres from the first trough, ad there is 3 metres betwee cosecutive troughs. stable had fills the troughs by carryig a bucket of water from the tap to each trough ad the returig to the tap. Thus she walks + 4 metres to fill the first trough, 10 metres to fill the secod trough, ad so o. (i) How far does the stable had walk to fill the k th trough? (ii) How far does the stable had walk to fill all troughs? (iii) The stable had walks 10 metres to fill all the troughs. How may water troughs are there? The School For Excellece 011 Trial Exam Preparatio Lectures Uit Maths Book 3 Page 37

18 QUESTION 9 (HSC 1998) fish farmer bega busiess o 1 Jauary 1998 with a stock of fish. He had a cotract to supply fish at a price of $10 per fish to a retailer i December each year. I the period betwee Jauary ad the harvest i December each year, the umber of fish icreases by 10 %. (i) Fid the umber of fish just after the secod harvest i December (ii) Show that F, the umber of fish just after the th harvest, is give by F (1.1). The School For Excellece 011 Trial Exam Preparatio Lectures Uit Maths Book 3 Page 38

19 (iii) Whe will the farmer have sold all his fish, ad what will his total icome be? The School For Excellece 011 Trial Exam Preparatio Lectures Uit Maths Book 3 Page 39

20 (iv) Each December the retailer offers to buy the farmer's busiess by payig $15 per fish for his etire stock. Whe should the farmer sell to maximise his total icome? The School For Excellece 011 Trial Exam Preparatio Lectures Uit Maths Book 3 Page 40