MattsMathLabs



-365760-236855 Sec 5.8 – Sec 6.2 – Mathematical Modeling (Arithmetic & Geometric Series) Name: -153670144780495123941156Carl Friedrich Gauss is probably one of the most noted complete mathematicians in history. As the story goes, he was potentially reconginized for his mathematical brilliance at the age of 8 when he was assigned busy work by his teacher for causing disruptions in class. He was told by the teacher to add all of the numbers between 1 and 100. 1 + 2 + 3 + 4 + 5 + 6 + …………..+ 97 + 98 +99 +100 = The teacher expected this task to take Guass several minutes to an hour to keep him busy but Gauss did it in seconds. So, the teacher thinking he had cheated told him to add the numbers between 1 and 200. This time Gauss didn’t even move, he just reponded with the answer. He had devised a trick to add consecutive numbers by pairing them in a special way at the age of 8. How did he do it?101 1 + 2 + 3 + 4 + 5 + …………+ 96 + 97 + 98 + 99 + 100 = 101101101101He determined that if you find the sum of the most outer pair of numbers it sums to 101 and that the next inner pair after that sums to 101 and so on. In short, there should be 50 pairs of numbers that sums to 101. So, this suggests: 1+2+3+4+5+6+………….+96+97+98+99+100=50×101=505050 pairs of 101 Using the techniqe that Gauss may have developed, determine the sum of all the integers from 1 to 200. 1+2+3+4+5+6+………….+196+197+198+199+200=It turns out that this strategy works for the partial sum of any Arithmetic Series. Consider writing it as a formula.5046094127551 Sn=n2a1+ anThe Sum of ‘n’ terms of an arithmetic seriesThe n2 represents the number of pairs of the terms that form the special sum.The “a1” represents the first term of the series.The “an” represents the last term of the series.Determine the sum of the following partial arithmetic series using the formula. 1. 2+4+6+8+……….+116+118+120=2. Find the S62 of the following series: 4+9+14+19+……….M. Winking Unit 6-2 page 1075686792-208112Determine the sum of the following partial arithmetic series using the formula. 3. 30+26+22+………+-102+-106=4. Find the S42, given that a1 = 6and a42 = 129 5. Find the S39 given that a1 = 6 and d = 66. Find the S34 given that a34 = 73 and d = 2. 7. Determine the value of 8. Determine the value of 57912001416059. Addison decides to try to save money in a jar at home. She decides to save $20 the first week of the year and each week she will increase the amount she saves by $5. So, on the second week she will save $25 and then on third week she will save an additional $30. This process would repeat for the whole year of 52 weeks. How much money should she have in the jar at the end of the year? M. Winking Unit 6-2 page 108There are also formulas that can be created to find the sum of a Geometric Series. First consider the following series.3 + 6 + 12 + 24 + 48 + 96 + 192 + 384 + 768 +1536 + 3072=This could also be re-written as:3+32+322+323+324+325+326+327+328+329+3210=11th term10th term5th term4th term9th term8th term6th term3rd term2nd term7th term1st termSo, any geometric series could be written as:Sn=a1+a1r+a1r2+a1r3+a1r4+…………..+a1rn-2+a1rn-1Consider multiplying both sides by a “-r” -r?Sn=-a1r-a1r2-a1r3-a1r4-a1r5-…………..-a1rn-1-a1rnNext, add the two series similar to how you use elimination in solving a system of equations. + Sn= a1 +a1r +a1r2+a1r3+a1r4+…………..+a1rn-2+a1rn-1-r?Sn=-a1r-a1r2-a1r3-a1r4-a1r5-…………..-a1rn-1- a1rnThis formula works for the partial sum of any Geometric Series. Arithmetic (addition)Geometric (multiplication)The “n” represents the number of sequential terms to be included in the sum.The “a1” represents the first term of the series. Sn=a11- rn1- rThe “r” represents the common ratio from one term to the next.The Sum of ‘n’ terms of an arithmetic seriesDetermine the sum of the following partial geometric series using the formula. 1. Find the S14 of the following series: 2+6+18+54+162+ ……….2. 3-6+12-24+……-98304+196608=M. Winking Unit 6-2 page 109Arithmetic (addition)Geometric (multiplication)Determine the sum of the following partial geometric series using the formula. 3. Determine the sum of the first 11 terms (S11) for a geometric series given the first term is 6 (a1=6) and the common ratio is 5 (r=5).4. Given the sum of the first 12 terms of a geometric sequences sum to 20475 and the common ratio is 2 (r=2), determine the first term (a1). 7. Determine the value of 8. Determine the value of 60553601517659. 72+36+18+9+4.5+……..= 10. 13+16+112+124+148+……..= 11. Determine the value of 12. Determine the value of M. Winking Unit 6-2 page 110Using the Algebra or the Infinite Geometric Series formulas determine the fraction for the following repeating decimals.13. 0.5555555555=14. 0.3434343434= 13. 14.14141414=14. 0.450450450450= 15. 0.99999999=57245255270516. Kelly decides to start saving money. On the first week of the year, she saves one cent ($0.01). Then, for each week that follows she continues to double the amount she saved the previous week. So, on the second week she saved an additional 2 cents ($0.02) and the 3rd week 4 cents ($0.04). If this process were able to be continued for the entire year of 52 weeks, how much money would Kelly have saved by the end of the year?354330044196016. Sarah Pinkski was creating a pattern using triangle tiles. She wanted to show each successive step to show how her pattern grows. She has already used 40 triangular tiles to create the pattern below. If she continued how many tiles would it take in total to create 10 steps of the design? M. Winking Unit 6-2 page 111 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download