Numerical sequences with computer

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1 Nauki ścisłe priorytetem społeczeństwa opartego a wiedzy Artykuły a platformę CMS S t r o a 1 Dr Aa Rybak Istitute of the Computer Sciece Uiversity i Białystok Numerical sequeces with computer Itroductio I the article will be preseted the way of usig spreadsheet for checkig the features of umeric sequeces (arithmetical ad geometrical), as well as for solvig problems described with the help of sequeces. Issues which exemplify the meaig of preseted problems, come from differet fields of sciece ad from the everyday life. The spreadsheet is well-kow for every user of the computer, but the use of it i the mathematics is rare, so the article will make the attempt to chage this situatio. Theoretical itroductio: Arithmetic sequece Numeric sequece (a ) is called arithmetical with differece of r, if a +1 = a +r for each N +. The real umber r is costat. The followig relatios are true: a = a 1 + (-1)r, S = a 1 + a a = a a 1 2a1 ( 1) r 2 2. S is called the -th partial sum of the sequece, whereas the sequece of partial sums (S ) is called the umerical series. How the computer ca help us examie arithmetic sequece - examples Example 1. Examie which of the preseted sequeces are arithmetic sequeces. What is the first term ad what is the commo differece? a) a 2 1, Projekt współfiasoway przez Uię Europejską w ramach Europejskiego Fuduszu Społeczego

2 Nauki ścisłe priorytetem społeczeństwa opartego a wiedzy Artykuły a platformę CMS S t r o a 2 b) b = 2 +1, c) u Aswerig this questio requires checkig, whether the differece of the -th term ad the previous oe is permaet idepedetly from (>1). Certaily you are able to desig the appropriate table which will carry it out: a =2/(+1 ) a +1 -a b =*+1 b +1 -b u =3-/3 u +1 -u 1 1, ,00 1 2, ,3333 0, ,00 3,00 2 2,3333-0, ,5000 0, ,00 5,00 3 2,0000-0, ,6000 0, ,00 7,00 4 1,6667-0, ,6667 0, ,00 9,00 5 1,3333-0, ,7143 0, ,00 11,00 6 1,0000-0, ,7500 0, ,00 13,00 7 0,6667-0, ,7778 0, ,00 15,00 8 0,3333-0, ,8000 0, ,00 17,00 9 0,0000-0, ,8182 0, ,00 19, ,3333-0, ,8333 0, ,00 21, ,6667-0, ,8462 0, ,00 23, ,0000-0, ,8571 0, ,00 25, ,3333-0, ,8667 0, ,00 27, ,6667-0,3333 Aalysis of the above give tables allows to euciate hypothesis : Sequece u is a arithmetic sequece, sequeces a ad b are't arithmetic sequeces. Example 2. Show that sequece (u ), which geeral expressio is defiited with the formula u = a* + b, a, b R is a arithmetic sequece. Projekt współfiasoway przez Uię Europejską w ramach Europejskiego Fuduszu Społeczego

3 Nauki ścisłe priorytetem społeczeństwa opartego a wiedzy Artykuły a platformę CMS S t r o a 3 Above tables ad graphs illustrate examiig the problem for specific rates a ad b. Chagig i the appropriate cells of the sheet the value of these rates you ca check the truth of the hypothesis costructed i the task for may differet a ad b. Remember that you should coduct formal evidece idepedetly of computer observatio. Example 3. Betwee the 1 of March ad the 31 of March surise o the 40 of the orth latitude takes place every day about 1,6 mi. earlier tha o the day before. What time will the su rise o the 21 of March, if o the 1 of March it rises at 6.33 a.m.? O which day the su will rise at 5.53 a.m.? data hour differece 01-III 06:33:00 00:01:36 02-III 06:31:24 Projekt współfiasoway przez Uię Europejską w ramach Europejskiego Fuduszu Społeczego

4 Nauki ścisłe priorytetem społeczeństwa opartego a wiedzy Artykuły a platformę CMS S t r o a 4 03-III 06:29:48 04-III 06:28:12 05-III 06:26:36 06-III 06:25:00 07-III 06:23:24 08-III 06:21:48 09-III 06:20:12 10-III 06:18:36 11-III 06:17:00 12-III 06:15:24 13-III 06:13:48 14-III 06:12:12 15-III 06:10:36 16-III 06:09:00 17-III 06:07:24 18-III 06:05:48 19-III 06:04:12 20-III 06:02:36 21-III 06:01:00 22-III 05:59:24 23-III 05:57:48 24-III 05:56:12 25-III 05:54:36 26-III 05:53:00 27-III 05:51:24 28-III 05:49:48 29-III 05:48:12 30-III 05:46:36 31-III 05:45:00 01-IV 05:43:24 I the above sheet the cells of the first colum were formatted so that it was possible to save the dates i them ad to perform operatios o their cotets accordig to features of the dates (e.g. chage of the moth i the right momet). I the uits of the secod colum are data like "time" ad thaks to the appropriate format it's possible to perform operatios icludig the features of the time (chage of the miute after 60 secods, chage of the hour after 60 miutes). It results from the task that hours o which the su is risig create arithmetic sequece with the differece of 1 mi ad 36 secods. As iitial values were iput ito the sheet: date 1. III (i the cell A2), 6:33 a.m. (i the cell B2) ad the commo differece of the sequece (cell C2 ). Calculatios were performed thaks to eterig formula: = A2 + 1 i cell A3, = B2 - $C$2 i cell B3 Projekt współfiasoway przez Uię Europejską w ramach Europejskiego Fuduszu Społeczego

5 Nauki ścisłe priorytetem społeczeństwa opartego a wiedzy Artykuły a platformę CMS S t r o a 5 ad copyig them ito cells located i the ext rows. From the table we ca see that o 21 March the su will rise at 6:01 ad at 5:53 the su will rise o the 26 of March. Example 4. I 1980 the populatio of people i a small tow rose by 4200 persos. Every year, i the ext decade, the populatio growth i this tow was reduced to 20 persos aually. How may people did the populatio umber i this tow rise from 1980 to 1990? Year Icrease of populatio total I the above sheet i order to estimate the total of populatio growth i was used fuctio SUM = SUM (F2: F12) i cell F14 The populatio growth i idividual years was estimated with the help of the formula takig features of arithmetic sequeces ito accout. Tasks for idepedet work: Task 1. Check, if showed sequeces are arithmetical. If yes, defie their mootoicity: a)a = 3 2, b)b = 3 2 2, c)c = 2-2. What methods ca you use to examie above preseted sequeces? Projekt współfiasoway przez Uię Europejską w ramach Europejskiego Fuduszu Społeczego

6 Nauki ścisłe priorytetem społeczeństwa opartego a wiedzy Artykuły a platformę CMS S t r o a 6 Task 2. Show that if sequece (a ) is arithmetic sequece, the all the poits with coordiates (; a ) belog to oe straight lie. Task 3. Ja Nowak purchased a buildig, i which he developed a eterprise. Every year he pays off the istalmet of credit cotracted for this purpose. The istalmet is PLN Activity brought o the first year the profit of PLN 70200, every followig year the profit rises by PLN After how may years the differece betwee the profit ad the height of istalmet will exceed PLN ? Task 4. While coductig botaical observatio was oticed that the diameter of the tree icreases for the same size every year. If the diameter was equal of 61 mm at the ed of the sixth year of the growth of the tree, ad 76 mm at the ed of the teth year, what was the diameter of the tree at the ed of the first year? Task5. As a result of measuremets ad proper computer simulatios was verified that directly after buildig the width of the Large Pyramid i Giza was reduced by 1.57 m for every metre of its height. O what height the width is equal to m, if the measured width at the level of 1 m is equal to m? How high was the pyramid at the begiig of its existece? Theoretical itroductio: Geometrical sequece We call umeric sequece (a ) geometrical with the commo ratio q differet from the zero, if a +1 = a *q for every N +. The real umber q is costat. The followig relatios are true: a = a 1 * q -1, 1 q S a1 1 q, if q 1, S = *a 1, if q=1. Projekt współfiasoway przez Uię Europejską w ramach Europejskiego Fuduszu Społeczego

7 Nauki ścisłe priorytetem społeczeństwa opartego a wiedzy Artykuły a platformę CMS S t r o a 7 Amog geometrical sequeces there are special sequeces, for which q <1. These are fast decreasig sequeces. The sum of their all terms is fiite. I this case geometrical series (S ) is coverget ad its sum is umber: a1 S 1 q How the computer ca help us i explorig geometrical sequeces examples Example 1. Check, which of the show sequeces are geometrical sequeces: a) a = 2, b) b = 2, c) c 3 4, d) d 1 Examiig above show sequeces cosists of checkig, whether the quotiet of the -th ad the previous elemet is fixed irregardless from (>1). a =2^ a +1 /a b = 2 b +1 /b c =3/4 c +1 /c d =/(- 1) d +1 /d , , ,00 2 0, , ,5000 0, ,25 3 0, , ,3333 0, ,78 4 0, , ,2500 0, ,56 5 0, , ,2000 0, ,44 6 0, , ,1667 0, ,36 7 0, , ,1429 0, ,31 8 0, , ,1250 0, ,27 9 0, , ,1111 0, , , ,2500 Aalysis of appropriate sheets allows to express hypothesis : Sequeces with geeral terms a ad c are geometrical sequeces, but sequeces with geeral terms b ad d are't. Example 2. Half-life of itroge 13 (the isotope of itroge) takes about 10 miutes. How much of itroge 13 will be i the laboratory at 4 p.m.,if at 3.10 p.m. the sample has the mass of 4 mg? Projekt współfiasoway przez Uię Europejską w ramach Europejskiego Fuduszu Społeczego

8 Nauki ścisłe priorytetem społeczeństwa opartego a wiedzy Artykuły a platformę CMS S t r o a 8 Half-life Time Weight of the sample i grams 00:10 15: : : :40 0,5 15:50 0,25 16:00 0,125 Example 3. A turtle is makig 2m log way o a straight lie withi 1 mi. Durig the ext miute it is coverig the distace of 1 m, i every ext miute it is makig half of the previous distace. If the turtle cotiues to walk forever- how far will it get? The umber of miutes sum of roads of the turtle way(i miutes) 1 2, , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , Projekt współfiasoway przez Uię Europejską w ramach Europejskiego Fuduszu Społeczego

9 Nauki ścisłe priorytetem społeczeństwa opartego a wiedzy Artykuły a platformę CMS S t r o a , , , , , , , , , , , , , , , , , , , , The coclusio from aalysis of the above sheet ca be surprisig for may people: walkig ay umber of miutes the turtle will ever exceed the distace of 4 metres. Tasks for idepedet work: Task 1. Examie the mootoicity of the geometrical sequece give with geeral formula: a) a = 2, b) b =3 (-1) -1, 3 c) c. 2 1 Task 2. Every ext year the value of a car costitutes 70% of its value from the previous year. Directly after producig the car cost PLN What will be its value after seve years? Task 3. Somebody made up a sesatioal rumour ad aouced it withi a hour to 10 persos. Next, each of this persos passed the rumour withi 10 miutes to 10 other persos, etc. After how may hours all the people livig o the Earth would kow the rumour? Assume that o the Earth live 7 * 10 9 people. Task 4 They built o a playgroud a system of objects to overcome by childre (feces, ladders, etc.) i the form of equilateral triagles iscribed i each other. The side of the biggest triagle has the legth of 32m. Iside was built ext triagle, as cusp we accept the cetres of sides of the bigger triagle. The operatio was repeated five times. Fid the legth of the side of the triagle formed as a result of the fourth performace of the operatio. How may metres of feces were used to build this structure? Task 5. Projekt współfiasoway przez Uię Europejską w ramach Europejskiego Fuduszu Społeczego

10 Nauki ścisłe priorytetem społeczeństwa opartego a wiedzy Artykuły a platformę CMS S t r o a 10 A tradesma sold a horse for PLN 165, but the buyer perceived himself, that he did't eed such a horse ad retured it to the ower with words: This horse is't worth the moey I paid for him. So the tradesma suggested other coditios: If you thik that the price of the horse is too high, buy oly stub ails which the horse has i the horseshoes, ad I will give you the horse for free. There are 6 stub ails i every horseshoe (i total 24). For the first oe you will pay ¼ of the pey, for the ext oe ½ of the pey, for the third 1 pey ad so o, for ay ext oe you will pay twice as much as for the previous oe. This way a miimum price eticed the villager. He accepted the coditio from the seller i the belief that he wo't pay more tha 2 zloty. Was he right? How much did he save or how much did lose? Projekt współfiasoway przez Uię Europejską w ramach Europejskiego Fuduszu Społeczego