= = =

Sec 5.8 Sec 6. Mathematical Modelig (Arithmetic & Geometric Series) Name: Carl Friedrich Gauss is probably oe of the most oted complete mathematicias i history. As the story goes, he was potetially recogiized for his mathematical brilliace at the age of 8 whe he was assiged busy work by his teacher for causig disruptios i class. He was told by the teacher to add all of the umbers betwee ad 00. + + 3 + 4 + 5 + 6 +..+ 97 + 98 +99 +00 = The teacher expected this task to take Guass several miutes to a hour to keep him busy but Gauss did it i secods. So, the teacher thikig he had cheated told him to add the umbers betwee ad 00. This time Gauss did t eve move, he just repoded with the aswer. He had devised a trick to add cosecutive umbers by pairig them i a special way at the age of 8. How did he do it? + + 3 + 4 + 5 + + 96 + 97 + 98 + 99 + 00 = 0 0 0 0 0 He determied that if you fid the sum of the most outer pair of umbers it sums to 0 ad that the ext ier pair after that sums to 0 ad so o. I short, there should be 50 pairs of umbers that sums to 0. So, this suggests: + + 3 + 4 + 5 + 6 +. +96 + 97 + 98 + 99 + 00 = 50 0 = 5050 Usig the techiqe that Gauss may have developed, determie the sum of all the itegers from to 00. + + 3 + 4 + 5 + 6 +. +96 + 97 + 98 + 99 + 00 = 50 pairs of 0 It turs out that this strategy works for the partial sum of ay Arithmetic Series. Cosider writig it as a formula. S = (a + a ) The Sum of terms of a arithmetic series The represets the umber of pairs of the terms that form the special sum. The a represets the first The a represets the last Determie the sum of the followig partial arithmetic series usig the formula.. + 4 + 6 + 8 +. +6 + 8 + 0 =. Fid the S 6 of the followig series: 4 + 9 + 4 + 9 +. M. Wikig Uit 6- page 07

Determie the sum of the followig partial arithmetic series usig the formula. 3. 30 + 6 + + + ( 0) + ( 06) = 4. Fid the S 4, give that a = 6 ad a 4 = 9 5. Fid the S 39 give that a = 6 ad d = 6 6. Fid the S 34 give that a 34 = 73 ad d =. 7. Determie the value of 3 8 8. Determie the value of 8 4 9. Addiso decides to try to save moey i a jar at home. She decides to save $0 the first week of the year ad each week she will icrease the amout she saves by $5. So, o the secod week she will save $5 ad the o third week she will save a additioal $30. This process would repeat for the whole year of 5 weeks. How much moey should she have i the jar at the ed of the year? M. Wikig Uit 6- page 08

There are also formulas that ca be created to fid the sum of a Geometric Series. First cosider the followig series. 3 + 6 + + 4 + 48 + 96 + 9 + 384 + 768 + 536 + 307 = This could also be re-writte as: 3 + 3() + 3( ) + 3( 3 ) + 3( 4 ) + 3( 5 ) + 3( 6 ) + 3( 7 ) + 3( 8 ) + 3( 9 ) + 3( 0 ) = st term d term 3 rd term 4 th term 5 th term 6 th term 7 th term 8 th term 9 th term 0 th term th term So, ay geometric series could be writte as: S = a + a (r) + a (r ) + a (r 3 ) + a (r 4 ) +.. +a (r ) + a (r ) Cosider multiplyig both sides by a r r S = a r a (r ) a (r 3 ) a (r 4 ) a (r 5 ).. a (r ) a (r ) Next, add the two series similar to how you use elimiatio i solvig a system of equatios. + S = a + a (r) + a (r ) + a (r 3 ) + a (r 4 ) +.. +a (r ) + a (r ) r S = a r a (r ) a (r 3 ) a (r 4 ) a (r 5 ).. a (r ) a (r ) This formula works for the partial sum of ay Geometric Series. The a represets the first The Sum of terms of a arithmetic series S = a ( r ) ( r) The represets the umber of sequetial terms to be icluded i the sum. The r represets the commo ratio from oe term to the ext. d a a a r a Arithmetic a a d S a a Geometric a a r a r S r Determie the sum of the followig partial geometric series usig the formula.. Fid the S 4 of the followig series: + 6 + 8 + 54 + 6 +.. 3 6 + 4 + 98304 + 96608 = M. Wikig Uit 6- page 09

Determie the sum of the followig partial geometric series usig the formula. 3. Determie the sum of the first terms (S ) for a geometric series give the first term is 6 (a =6) ad the commo ratio is 5 (r=5). 4. Give the sum of the first terms of a geometric sequeces sum to 0475 ad the commo ratio is (r=), determie the first term (a ). d a a a r a Arithmetic a a d S a a Geometric a a r a r S r 5 7. Determie the value of 4 8. Determie the value of 9. 7 + 36 + 8 + 9 + 4.5 +.. = 0. 3 + 6 + + 4 + 48 +.. =. Determie the value of. Determie the value of 3 4 M. Wikig Uit 6- page 0

Usig the Algebra or the Ifiite Geometric Series formulas determie the fractio for the followig repeatig decimals. 3. 0.5555555555 = 4. 0.3434343434 = 3. 4.4444 = 4. 0.450450450450 = 5. 0.99999999 = 6. Kelly decides to start savig moey. O the first week of the year, she saves oe cet ($0.0). The, for each week that follows she cotiues to double the amout she saved the previous week. So, o the secod week she saved a additioal cets ($0.0) ad the 3 rd week 4 cets ($0.04). If this process were able to be cotiued for the etire year of 5 weeks, how much moey would Kelly have saved by the ed of the year? 6. Sarah Pikski was creatig a patter usig triagle tiles. She wated to show each successive step to show how her patter grows. She has already used 40 triagular tiles to create the patter below. If she cotiued how may tiles would it take i total to create 0 steps of the desig? M. Wikig Uit 6- page