Intro to Sequences / Arithmetic Sequences and Series Levels
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1 Itro to Sequeces / Arithmetic Sequeces ad Series Levels Level : pg. 569: #7, 0, 33 Pg. 575: #, 7, 8 Pg. 584: #8, 9, 34, 36 Levels, 3, ad 4(Fiboacci Sequece Extesio) See Hadout Check for Uderstadig
2 Level 9., 9., ad 9.4 Aswers Pg. 569: 7. 5, 8,, 4, 7, 0 0., 5, 0, 7, 6, Pg. 575: Yes, 0 8. No Pg. 59: Sequece; ifiite 36. Series; fiite Level 9., 9., ad 9.4 Aswers Pg. 569: 7. 5, 8,, 4, 7, 0 0., 5, 0, 7, 6, Pg. 575: Yes, 0 8. No Pg. 59: Sequece; ifiite 36. Series; fiite Level 9., 9., ad 9.4 Aswers Pg. 569: 7. 5, 8,, 4, 7, 0 0., 5, 0, 7, 6, Pg. 575: Yes, 0 8. No Pg. 59: Sequece; ifiite 36. Series; fiite
3 Level 9., 9., ad 9.4. Write the first six terms of the sequece. a 4. Write the ext term i the sequece, The, write a rule for the th term. 6, 9,, 5,... 3 Write a rule for the th term of the arithmetic sequece. d 4, a 7 4. Write a rule for the th term of the arithmetic sequece. d 3, a Fid the sum of the first 0 terms usig the formula give Level 9., 9., ad 9.4. Write the first six terms of the sequece. a 4. Write the ext term i the sequece, The, write a rule for the th term. 6, 9,, 5,... 3 Write a rule for the th term of the arithmetic sequece. d 4, a 7 4. Write a rule for the th term of the arithmetic sequece. d 3, a Fid the sum of the first 0 terms usig the formula give
4 Level 9., 9., ad 9.4 Aswers. Write the first six terms of the sequece. a 4 Aswer: ,3,,,, Write the ext term i the sequece, The, write a rule for the th term. 6, 9,, 5,... Aswer: 8; a Write a rule for the th term of the arithmetic sequece. d 4, a 7 Aswer: a 4 4. Write a rule for the th term of the arithmetic sequece. d 3, a8 5 Aswer: a Fid the sum of the first 0 terms usig the formula give Aswer: 344 Level 9., 9., ad 9.4 Aswers. Write the first six terms of the sequece. a 4 Aswer: ,3,,,, Write the ext term i the sequece, The, write a rule for the th term. 6, 9,, 5,... Aswer: 8; a Write a rule for the th term of the arithmetic sequece. d 4, a 7 Aswer: a 4 4. Write a rule for the th term of the arithmetic sequece. d 3, a8 5 Aswer: a Fid the sum of the first 0 terms usig the formula give Aswer: 30
5 Level 3 9., 9., ad 9.4. Graph the arithmetic sequece a 3 for the first six terms.. Write a rule for the th term of the arithmetic sequece a6 7., a Two terms of a arithmetic sequece are a8 ad a5 30. Fid the value of for which a A. Fid the sum of the first 8 terms of the arithmetic sequece 6 6 B. Fid for the give sum S. S 44 Level 3 9., 9., ad 9.4. Graph the arithmetic sequece a 3 for the first six terms.. Write a rule for the th term of the arithmetic sequece a6 7., a Two terms of a arithmetic sequece are a8 ad a5 30. Fid the value of for which a A. Fid the sum of the first 8 terms of the arithmetic sequece 6 6 B. Fid for the give sum S. S 44
6 Level 3 9., 9., ad 9.4 Aswers. Graph the arithmetic sequece a 3 for the first six terms. Aswer:. Write a rule for the th term of the arithmetic sequece a6 7., a3 44 Aswer: a Two terms of a arithmetic sequece are a8 ad a5 30. Fid the value of for which a 50. Aswer: 3 4. A. Fid the sum of the first terms of the arithmetic sequece 6 6 Aswer: 344 B. Fid for the give sum S. S 44 Aswer: 7 Level 3 9., 9., ad 9.4 Aswers. Graph the arithmetic sequece a 3 for the first six terms. Aswer:. Write a rule for the th term of the arithmetic sequece a6 7., a3 44 Aswer: a Two terms of a arithmetic sequece are a8 ad a5 30. Fid the value of for which a 50. Aswer: 3 4. A. Fid the sum of the first terms of the arithmetic sequece 6 6 Aswer: 344 B. Fid for the give sum S. S 44 Aswer: 7
7 Level 4 Fiboacci Sequece Extesio Oe famous mathematical sequece is the Fiboacci sequece. You ca fid each term of the sequece usig additio, but the sequece is ot arithmetic. The Fiboacci sequece is a recursive sequece that we will lear about later. The sequece is as follows:,,, 3, 5, 8, 3,.... Fid diagoals i Pascal s Triagle at the right by startig with the first i each row ad movig oe row up ad oe umber to the right. For example, the diagoal startig i the fifth row is, 3,. The diagoal startig i the sixth row is, 4, 3. For each diagoal, write the sum of its etries. What patter do the sums form?. A. Geerate the first te terms of the Fiboacci sequece. B. Fid the sum of the first te terms of the Fiboacci sequece. Divide the sum by. What do you otice? C. Choose two umbers other tha ad. Geerate a Fiboacci-like sequece from them. Write the first te terms of your sequece, fid the sum, ad divide the sum by. What do you otice? D. Make a Cojecture. What is the sum of the first te terms of ay Fiboacci-like sequece? 3. A. Study the patter at the right. Write the ext lie. B. Without calculatig, use the patter to predict the sum of the squares of the first te terms of the Fiboacci sequece. C. Verify the predictio you made i part B. 4. If you divide each Fiboacci umber by the immediately precedig oe, the sequece of ratios you get approaches a umber called. Fid by usig the first 0 terms. What is ad what is aother ame for ( you may ot kow the alterative ame)? 5. Usig the first te terms of the Fiboacci sequece, try the followig experimet: Pick ay Fiboacci umber. Take the two umbers o either side of the oe you picked, multiply them together, ad subtract the square of the umber you picked. What do you otice? Try this with several differet umbers. Ca you predict the aswer before you do the arithmetic? A explicit rule for the Fiboacci sequece is give by a 5. Show the work ad use your calculator to evaluate this formula for = ad = 3. Write out the umbers i the Fiboacci sequece to get the th ad 3 th term after you have used the formula to cofirm that the formula worked.
8 7. Exted the sequece backwards ad see if you ca explai the patter. 8. List the first 5 umbers i the Fiboacci sequece. Do fiite differeces for six times ad look for the patter. What do you fid? 9. Cosider ay three cosecutive umbers i the Fiboacci sequece. Square the middle umber ad compare it to the product of the remaiig two umbers. Repeat this umerous times. Make a cojecture about the patter that you fid. 0. Cosider ay four cosecutive umbers i the Fiboacci sequece. Multiply the two ier umbers ad compare it to the product of the two outer umbers. Repeat this umerous times. Make a cojecture about the patter that you fid.. Experimet i a similar way with five or six or more cosecutive Fiboacci umbers to discover other patters. Write what patters you fid.
9 Level 4 Fiboacci Sequece Extesio Aswers Oe famous mathematical sequece is the Fiboacci sequece. You ca fid each term of the sequece usig additio, but the sequece is ot arithmetic. The Fiboacci sequece is a recursive sequece that we will lear about later. The sequece is as follows:,,, 3, 5, 8, 3,.... Fid diagoals i Pascal s Triagle at the right by startig with the first i each row ad movig oe row up ad oe umber to the right. For example, the diagoal startig i the fifth row is, 3,. The diagoal startig i the sixth row is, 4, 3. For each diagoal, write the sum of its etries. What patter do the sums form? Aswer: st row diagoal: ; d row diagoal: ; 3 rd row diagoal:, 4 th row diagoal: 3; 5 th row diagoal: 5; 6 th row diagoal: 8; 7 th row diagoal: 3; The sums replicate the Fiboacci Sequece.. A. Geerate the first te terms of the Fiboacci sequece. Aswer:,,, 3, 5, 8, 3,, 34, 55 B. Fid the sum of the first te terms of the Fiboacci sequece. Divide the sum by. What do you otice? Aswer: 43; 3; the aswer is the seveth term C. Choose two umbers other tha ad. Geerate a Fiboacci-like sequece from them. Write the first te terms of your sequece, fid the sum, ad divide the sum by. What do you otice? Aswer: the aswer is the seveth term D. Make a Cojecture. What is the sum of the first te terms of ay Fiboacci-like sequece? Aswer: it will be times the seveth term 3. A. Study the patter at the right. Write the ext lie. Aswer: B. Without calculatig, use the patter to predict the sum of the squares of the first te terms of the Fiboacci sequece. Aswer: 4895 C. Verify the predictio you made i part B. Aswer: If you divide each Fiboacci umber by the immediately precedig oe, the sequece of ratios you get approaches a umber called. Fid by usig the first 0 terms. What is ad what is aother ame for ( you may ot kow the alterative ame)? Aswer: You should get but if you cotiue you will fid that ad is called the golde ratio.
10 5. Usig the first te terms of the Fiboacci sequece, try the followig experimet: Pick ay Fiboacci umber. Take the two umbers o either side of the oe you picked, multiply them together, ad subtract the square of the umber you picked. What do you otice? Try this with several differet umbers. Ca you predict the aswer before you do the arithmetic? Aswer: The product mius the square is always or -, ad these values alterate, so if you started with the th Fiboacci umber, you ll get A explicit rule for the Fiboacci sequece is give by a. Show the work ad use your calculator to evaluate this 5 formula for = ad = 3. Write out the umbers i the Fiboacci sequece to get the th ad 3 th term after you have used the formula to cofirm that the formula worked. Aswer: 44, Exted the sequece backwards ad see if you ca explai the patter. Aswer:...,-8, 5, -3,, -,, 0 the sigs of the umbers will alterate betwee etc. 8. List the first 5 umbers i the Fiboacci sequece. Do fiite differeces for six times ad look for the patter. What do you fid? Aswer: You will obtai the Fiboacci sequece goig i both directios. The umbers to the right of 0 are positive while the umbers to the left of 0 will alterate sigs but are still the Fiboacci sequece. 9. Cosider ay three cosecutive umbers i the Fiboacci sequece. Square the middle umber ad compare it to the product of the remaiig two umbers. Repeat this umerous times. Make a cojecture about the patter that you fid. Aswer: The product of the outer umbers is alterately oe more or oe less tha the square of the middle umber. 0. Cosider ay four cosecutive umbers i the Fiboacci sequece. Multiply the two ier umbers ad compare it to the product of the two outer umbers. Repeat this umerous times. Make a cojecture about the patter that you fid. Aswer: The product of the outer umbers is alterately oe more or oe less tha the product of the ier umbers.. Experimet i a similar way with five or six or more cosecutive Fiboacci umbers to discover other patters. Write what patters you fid.
11 Check for Uderstadig Aswers. Write a rule for the th term of the sequece,, 0, 9, 38,.... The fid a. 5 Aswer: a 9 7; a5 8. Fid the sum of the first 50 terms of the arithmetic series Aswer: 385
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