Overview I this activity, studets will lear how to costruct a.ts file to ivestigate sequeces ad series ad to discover some iterestig patters while avoidig tedious calculatios. They will explore both arithmetic ad geometric sequeces ad series as well as the Fiboacci Sequece. Materials TI-Nspire hadheld or Computer Software This documet is ot meat to be completed i a sigle class period. Pick ad choose the material to use as you cover the topics. Note: The istructios provided assume the documet is created o a hadheld. Teacher Software istructios are icluded if they differ from the hadheld. Part Preparig the documet. Hadheld: Press c > New Documet. Software: Click File > New.. Select Add Notes. 3. Create a title page for the activity, if desired, by typig Buildig Sequeces ad Series with a Spreadsheet. Note: To obtai capital letters o the hadheld, press the g key, the the letter. Part Creatig the Spreadsheet 4. Hadheld: Press /~ > Add Lists & Spreadsheet (or press /I > Add Lists & Spreadsheet). Software: Click Isert > Lists & Spreadsheet. 5. Type the umeral ito cell A, ad press. Type the umeral 5 ito cell A, ad press. Istead of cotiuig to maually eter additioal umbers, you ca fill the spreadsheet automatically ( st Method): Hadheld: Positio the cursor i cell A. Press ad hold g ad press the dow arrow ( ) to highlight cells A ad A. Select MENU > Data > Fill. A dotted border is show aroud the two highlighted cells. Press the dow arrow ( ) util you reach cell A4. Press to fill the cells. Software: Highlight A ad A. Place your cursor i the bottom right had corer of cell A ad drag to cell A4. Alteratively, you ca highlight A ad A, right-click, ad select Fill. Move your cursor dow to cell A4, ad press eter o your keyboard. Sice each successive etry is 3 more tha the previous oe, we have created a arithmetic sequece. 0 Texas Istrumets Icorporated educatio.ti.com
Tech Tip: To jump up to the first etry i a list, press /7. To jump dow to the last etry i a list, press /. 6. Before eterig a formula, we will first clear the lists: Hadheld: Clear the list i Colum A by movig up ( ) to the top of Colum A util the etire colum is highlighted. Press. to delete the data from the colum. Software: Highlight Colum A usig the mouse, ad press the delete key o the computer keyboard. Tech Tip: If you are havig difficulty highlightig the colum o the hadheld, you probably did ot go up far eough. Whe you reach the cell that has the letter A, arrow up oe more time to highlight the etire colum. 7. ( d Method) Now that the lists are clear, re-eter the lists by usig a formula: Hadheld: Type the umeral ito cell A, ad press. Istead of automatically fillig the colum, we will create the same sequece recursively by utilizig a formula. I cell A, type a 3, ad press. Arrow up ( ) to select cell A. Select MENU > Data > Fill. Press the dow arrow ( ) util you reach cell A4. Press to fill the cells. Software: After eterig the formula, select cell A, place your cursor i the bottom right had corer of cell A, ad drag to cell A4 (or right-click i cell A, copy, drag your cursor to cell A4, right-click ad paste). Teacher Tip: By scrollig up ad dow Colum A, studets ca see how each term was geerated. For example, If cell A3 is selected, the bottom of the scree displays the formula, A3 a 3. Guide studets to utilize this iformatio to obtai a formula for fidig the th term of a arithmetic sequece. For example, lookig at the formula A3 a 3, we have: A3 a 3 a 3 3 a 3. Sice A4 a33, we have A4 a3 3 a3 3 a 3 3 3 a 333 a 3(3). We ca geeralize our results to obtai the formula. For, the of a arithmetic sequece is a a d, where th term a is the first term 0 Texas Istrumets Icorporated educatio.ti.com
ad d is the differece betwee cosecutive terms. 8. Use Colum B to sum the etries from Colum A. Hadheld: Type the umeral ito cell B, ad press. I cell B, type b a, ad press. Arrow up ( ) to select cell B. Select MENU > Data >Fill. Press the dow arrow ( ) util you reach cell B4. Press to fill the cells. Software: After eterig the formula, select cell B, place your cursor i the bottom right had corer of cell B, ad drag to cell B4. We see that the sum a a a3 a4 6. Teacher Tip: You might wat to guide studets to geerate a formula for a arithmetic series, the sum of the arithmetic sequece. For example, we ca sum the first four terms by simply addig them, 58 6. However, otice that both sums a a4 ad a a3 3, givig the total of 6. We ca look at additioal terms to see if this patter cotiues. Go back to Colum A ad fill dow to A8 usig the method i step 8 above. We see that a a8 a a7 a3 a6 a4 a5 5, four sums of 5, for a total of 00. Go back to Colum B, ad fill it dow to B8 to verify our results. To geeralize a formula for the sum of a arithmetic sequece, we will first assume that is eve. The sum, S, of the first terms is: S a a a a a a a a a pairs a a. A similar derivatio shows that this formula also holds for odd values of You might wat to tell studets about the story of the great mathematicia Carl Friedrich Gauss who, as a youg boy, was said to have bee asked by his teacher to sum the umbers from to 00 ad did so almost immediately. Although, accordig to E.T. Bell i The Me of Mathematics, p. (New York: Simo ad Schuster, 937), the series was a more complicated arithmetic series, it still makes a good poit. 0 Texas Istrumets Icorporated 3 educatio.ti.com
. A alterative derivatio ca be obtaied by reversig the order of the terms i the series ad addig the two, as follows: S a a a a S a a a a S a a S a a S a a a d. Thus, The first versio of the above formula is useful whe we kow the first ad last terms of the sequece. The secod formula is useful whe the last term is ot kow Part 3 Geometric Sequeces ad Series 9. Hadheld: Select ~ > Isert > Problem > Add Lists & Spreadsheets. Software: Click Isert > Problem > Add Lists & Spreadsheets. Type the umeral 3 ito cell A, ad press eter. 0. We will create a geometric sequece by multiplyig each term by. Hadheld: I cell A, type a, ad press. Arrow up ( ) to select cell A. Select MENU > Data > Fill. Press the dow arrow ( ) util you reach cell A4. Press to fill the cells. Software: After eterig the formula, select cell A, place your cursor i the bottom right had corer of cell A, ad drag to cell A4. Teacher Tip: By scrollig up ad dow Colum A, studets ca see how each term was geerated. For example, If cell A3 is selected, the bottom of the scree displays the formula, A3 a. Guide studets to utilize this iformatio to obtai a formula for fidig the th term of a geometric sequece. For example, lookig at the formula A3 a, we have: 0 Texas Istrumets Icorporated 4 educatio.ti.com
A3 a a a. Sice A4 a3, we have 3 A4 a3 a a a a. the We ca geeralize our results to obtai the formula, for, th term of geometric sequece is a ad r is the ratio of successive terms. a r where, a is the first term. Use Colum B to sum the etries from Colum A. Hadheld: Type the umeral 3 ito cell B, ad press. I cell B type, b a, ad press. Arrow up ( ) to select cell B. Select MENU > Data >Fill. Press the dow arrow ( ) util you reach cell B4. Press to fill the cells. Software: After eterig the formula, select cell B, place your cursor i the bottom right had corer of cell B, ad drag to cell B4. We see that the sum a a a3 a4 45. Teacher Tip: You might wat to guide studets to geerate a formula for a geometric series, the sum of the geometric sequece. For example, we ca sum the first four terms, 3 6 4 45. We ca rewrite this as 3 S4 3 6 4 3 3 3 3 45. To geeralize a formula for the sum of a geometric sequece, we replace our costats with variables ad use our algebra skills. The sum, S, of the first terms is: S a a a Multiply both sides by r, ad add a. a a r a r a r a r 3 rs a r a a r a r a r a r a 3 a r a r a r a r a r a 3 4 Re-arrage the terms o the right side of the equatio, ad simplify: 0 Texas Istrumets Icorporated 5 educatio.ti.com
rs a a r a r a r a r a r a 3 4 a a r a r a r a r a r a r 3 4 S S a r Solve for S : rs a S a r rs S a r a a r S r a r S r We ca multiply both the umerator ad deomiator by - to obtai: a S a a r a r a r a r 3 r r, for r. Part 4 The Fiboacci Sequece The Fiboacci sequece is a well-kow recursively defied sequece that produces some iterestig relatioships. It is geerated by choosig values for the first two terms. The third term is obtaied by addig the first ad secod terms. The fourth term is the sum of the secod ad third terms. Each successive term is the sum of the previous two terms. Symbolically, F F. F I his 0 book, Liber Abaci, Leoardo of Pisa, who wrote uder the ame Fiboacci, itroduced the sequece to Wester Europea mathematics. The Liber Abaci bega the sequece with F F, ad we will begi our sequece the same way.. Hadheld: Select ~ > Isert > Problem > Add Lists & Spreadsheets. Software: Click Isert >Problem > Add Lists & Spreadsheets. 3. Hadheld: Type the umeral ito cell A, ad press. Type the umeral ito cell A, ad press. I A3, type a a, ad press. Arrow up ( ) to select cell A3. Select MENU > Data > Fill. Press the dow arrow ( ) util you reach cell A8. Press to fill the cells. http://e.wikipedia.org/wiki/fiboacci_sequece 0 Texas Istrumets Icorporated 6 educatio.ti.com
Software: After eterig the umeral ito cells A ad A, eter the formula a a ito cell A3, select cell A3, place your cursor i the bottom right had corer of cell A3, ad drag to cell A8. We will use Colum B to examie the ratios betwee cosective terms i Colum A. 4. Hadheld: I cell B, type a / a, ad press. Arrow up ( ) to select cell B. Select MENU > Data >Fill. Press the dow arrow ( ) util you reach cell B8. Press to fill the cells. Software: After eterig the formula, select cell B, place your cursor i the bottom right had corer of cell B, ad drag to cell B8. Tech Tip: To see decimal approximatios istead of fractios i Colum B, add a decimal poit to the etry i cell A, that is, [.]. To examie the ratios more closely, exted both colums to 5 terms. 5. Hadheld: I Colum A, select cell A8, ad select MENU > Data >Fill. Press the dow arrow ( ) util you reach cell A5. Repeat the same steps for Colum B. Software: I Colum A, select cell A8, place your cursor i the bottom right had corer of cell A8, ad drag to cell A5. Repeat for Colum B. Note: As the value of icreases, the ratio betwee cosecutive terms approaches the Golde Ratio, phi.6803398875. Teacher Tip: Note that the values of cosecutive ratios alterate betwee beig larger tha phi ad smaller tha phi. 6. Chage the values i cells A ad A to ay o-zero values. Note that as you chage these two values all of the other related values i the spreadsheet chage. Also, ote that the ratio betwee cosecutive terms always approaches phi. 0 Texas Istrumets Icorporated 7 educatio.ti.com
Teacher Tip: To compute the value of phi, we use the fact that F F F. Dividig both sides of the equatio by F, gives Thus, F F F F. F F F F F F Assumig the ratio of cosecutive terms approaches, the. Thus, 4 5 4 5 4 5 We discard the egative solutio to obtai 5.6803398875... Note: If you have a CAS versio of TI Nspire hadheld or computer software, you might wat to isert some symbolic etries to see the patter. For example, eter a ito cell A ad b ito cell A. Type the formula =a+a ito cell A3. 0 Texas Istrumets Icorporated 8 educatio.ti.com
Use Colum C to examie the ratios betwee every other term i Colum A. 7. Hadheld: I cell C3, type a3/ a, ad press. Arrow up ( ) to select cell C3. Select MENU > Data >Fill. Press the dow arrow ( ) util you reach cell C5. Press to fill the cells. Software: After eterig the formula, select cell C3, place your cursor i the bottom right had corer of cell C3, ad drag to cell C5. Note that the sum approaches phi + Teacher Tip: It is iterestig to ote that, as show below, 5 5 5 4 6 5 4 4 5 4 4 5 You might also wat to have studets geerate the ratio,, ad subsequet ratios to see that the ratios approach 3.. There are may other iterestig relatioships amog the terms of the Fiboacci sequece. 3 The use of a spreadsheet will make it possible for studets to easily explore them. 3 See Fasciatig Fiboaccis, Mystery ad Magic i Numbers by Trudi Hammel Garlad (Palo Alto, CA: Dale Seymour Publicatios, 987) ad Spreadsheets ad the Fiboacci Sequece by Kyle Schultz, http://jwilso.coe.uga.edu/emat6680fa05/schultz/assigmet/fiboacci 0 Texas Istrumets Icorporated 9 educatio.ti.com