1 MCR3U Uit 8: Sequeces & Series Page 1 of Arithmetic Sequeces Defiitio: A sequece is a comma separated list of ordered terms that follow a patter. Examples: 1, 2, 3, 4, 5 : a sequece of the first 5 itegers. 1, 4, 9, 16, : a sequece of the perfect squares startig at 1. The at the ed of the sequece is called the ellipsis ad idicates that the sequece cotiues idefiitely ,,,, : a sequece of fractios where deomiators are the itegers from Q1. Idetify the patter i each of the followig sequeces, ad the write the ext 2 terms: 1, 4, 7, 10, 13, 2, 4, 8, 16, ,,,, Formulas ad Notatio We use the followig letters ad otatio to represet iformatio about a sequece: : represets the term umber. The term umber must be a iteger startig at 1. t : represets the value of the th term. It is sometimes writte as t() which follows fuctio otatio. For example i the sequece 1, 4, 7, 10 the 7 is the 3 rd term, so we ca say that t 3 = 7. Notice that the subscript of 3 idicates that it is the third term. We ca use this otatio to defie formulas for sequeces based o the value of. Q2. Write the first three terms of each sequece, give the formula. a) t = b) t() = 2
2 MCR3U Uit 8: Sequeces & Series Page 2 of 2 Defiitio: A arithmetic sequece is ay sequece that shows a commo differece betwee terms. That is, the differece betwee ay two cosecutive terms i the sequece is a costat value. Also, if you kow a term i the sequece, you ca add this commo differece to it to determie the value of the ext term. Q3. Determie if each sequece is arithmetic, ad if it is, write the ext two terms. a) 4, 9, 14, 19, 24,, b) 10, 7, 4, 1, -2,, c) 2, 5, 8, 11, 15,, Formula for Arithmetic Sequeces Let a represet the first term of the sequece. (first letter of alphabet first term) Let d represet the commo differece for arithmetic sequeces. (d for differece) Write expressios for the first 4 terms of a arithmetic sequece, usig a ad d. t 1 = t 2 = t 3 = t 4 = Use ay patters you see above to write a formula for the th term t, usig a,, & d. t = Q4. For each of the followig sequeces, idetify the formula for the th term ad use your formula to determie the 100 th term i the sequece. a) 5, 7, 9, 11, b) 10, 7, 4, 1, c),,,,
3 MCR3U Uit 8: Sequeces & Series Page 3 of 3 Q5. Determie the umber of terms i the sequece 17, 14, 11, 8,, 136. Q6. I a arithmetic sequece, t 4 = 3 ad t 15 = 19. Determie the first three terms. Q7. Determie the value of x so that the three give terms form a arithmetic sequece (x 2), (x 4), (2x + 1).
4 MCR3U Uit 8: Sequeces & Series Page 4 of Geometric Sequeces Warmup Q1. Idetify the followig sequeces as arithmetic or o-arithmetic. If the sequece follows a patter, write the ext three terms. a) 1, 4, 9, 16,,, b) 2, 6, 10, 14,,, c) 3, 6, 12, 24,,, Q2. I a arithmetic sequece, t = 20 ad t 10 = 32. Determie: a) The first term ad commo differece. 6 b) The formula for the th term of the sequece. c) The value of t 100
5 MCR3U Uit 8: Sequeces & Series Page 5 of 5 Defiitio: A geometric sequece is ay sequece that shows a commo ratio betwee terms. That is, the ratio betwee ay two cosecutive terms i the sequece is a costat value. Also, if you kow a term i the sequece, you ca multiply the term by the commo ratio to determie the value of the ext term. Q1. Determie if each sequece is geometric, ad if it is, write the ext two terms. a) 2, 6, 18, 54,, b) 1024, 512, 256, 128,, c) 8, 16, 32, 64, 126,, Formula for Geometric Sequeces Let a represet the first term of the sequece. (first letter of alphabet first term) Let r represet the commo ratio for geometric sequeces. (r for ratio) Write expressios for the first 4 terms of a geometric sequece, usig a ad r. t 1 = t 2 = t 3 = t 4 = Use ay patters you see above to write a formula for the th term t, usig a,, & r. t = Q2. For each of the followig sequeces, idetify the formula for the th term ad use your formula to determie the 10 th term i the sequece. a) 2, 6, 18, b) 1024, 512, 256, c) 2, -2, 2, -2,
6 MCR3U Uit 8: Sequeces & Series Page 6 of 6 3 Q3. Determie the umber of terms i the sequece 48, 24, 12,,. 64 Q4. I a geometric sequece, t 5 = 405 ad t 9 = Determie the first three terms. Q5. Determie the value(s) of x so that the three give terms form a geometric sequece. (x 2), (2x 4), (x + 1).
7 MCR3U Uit 8: Sequeces & Series Page 7 of Recursio Formulas I previous lessos, we have determied sequeces usig a formula for the th term. A example is the formula t = 2 + 3, which determies the arithmetic sequece 5, 7, 9, 11, Aother example is t = 2 1, which determies the geometric sequece 1, 2, 4, 8, 16, Such formulas are kow as explicit formulas. They ca be used to calculate ay term i a sequece without kowig the previous term. For example, the teth term i the sequece determied by the formula t = is: t 10 = 2(10) + 3, which equals 23. It is sometimes more coveiet to calculate a term i a sequece from oe or more previous terms i the sequece. Formulas that ca be used to do this are called recursio formulas. A recursio formula cosists of at least two parts. The parts give the value(s) of the first term(s) i the sequece, ad a equatio that ca be used to calculate each of the other terms from the term(s) before it. A example is the formula: t 1 = 5 t = t The first part of the formula shows that the first term is 5. The secod part of the formula shows that each term after the first term is foud by addig 2 to the previous term. The sequece geerated will be 5, 7, 9, 11, 13, which happes to be a arithmetic sequece with explicit formula t = Q1. Write the first 5 terms for the sequece defied by: t t 1 = 1 = t 1 + Q2. Is the sequece i Q1 above arithmetic or geometric? Recursio Formulas for Arithmetic ad Geometric Sequeces Arithmetic Sequeces t = a t 1 = t 1 + d Geometric Sequeces t = a t 1 = t 1 r
8 MCR3U Uit 8: Sequeces & Series Page 8 of 8 Fiboacci Sequece Oe of the most famous recursive sequeces is the Fiboacci Sequece, which was discovered by a Italia mathematicia amed Leoardo Fiboacci (c ). It is defied as: t1 = 1 t2 = 1 t = t + 1 t 2 This says that: the first term is 1, the secod term is 1, ad the every term afterwards is the sum of the previous two terms. Q3. Write the first 12 terms of the Fiboacci sequece. Q4. Is the Fiboacci sequece arithmetic or geometric? Iterestig Iformatio about the Fiboacci Sequece While the Fiboacci sequece is ot geometric, there is a patter betwee cosecutive terms. As the sequece cotiues idefiitely, the ratio betwee cosecutive terms approaches a value of approximately (Try it with the umbers foud i Q3) This value is kow as the golde ratio ad has a exact value of This umber ca also be foud usig ifiite patters ad The golde ratio is the oly positive aswer to the questio What umber mius oe is equal to the reciprocal of the umber? The golde ratio has a lot of applicatios i mathematics, architecture, art, ad ature
9 MCR3U Uit 8: Sequeces & Series Page 9 of 9 Q5. Each stroke of a vacuum pump removes oe third of the air remaiig i the cotaier. What percet of the origial quatity of air remais i the cotaier after 10 strokes, to the earest percet? Summary of Sequeces Sequeces Terms t value of th term term umber i sequece a first term of sequece d commo differece r commo ratio Formulas Arithmetic Sequeces t = a + ( 1)d Geometric Sequeces t = a(r) -1
10 MCR3U Uit 8: Sequeces & Series Page 10 of Arithmetic Series Defiitio: A series is the sum of terms i a sequece. A arithmetic series is the sum of terms i a arithmetic sequece Notatio: S represets the sum of the first terms i a sequece. (S for Sum) Usig Gauss Method to fid the Sum ie. S = t 1 + t 2 + t t -1 + t Q1. Determie the sum of the first 100 atural umbers, usig the Gauss Method. Gauss Method rewrite the series with the terms reversed. add the two series together look for patters to simplify, the rearrage for S S 100 = Gauss Method usig Geeral Terms Q2. Use the Gauss Method, to determie the sum of the first terms of a arithmetic sequece. [Hit: keep t 1 ad t, but rewrite the other terms usig t 1, t, ad d.] S = t 1 + t 2 + t t -2 + t -1 + t
11 MCR3U Uit 8: Sequeces & Series Page 11 of 11 Formulas for Arithmetic Series S = ( t1 + t ) 2 or by replacig t 1 = a, ad t = a+(-1)d, we get = [ 2a + ( 1) d] S 2 The versio of the formula you use depeds o what iformatio you have. If you have t, use the first versio. If you have d, use the secod versio. Notice tha i either equatio, you must kow. Q3. Determie the sum of the first 150 terms of the arithmetic series Q4. Determie the sum of the first 50 terms of a arithmetic sequece defied by t = 4 1. Q5. Determie the sum of the arithmetic series:
12 MCR3U Uit 8: Sequeces & Series Page 12 of 12 Q6. The first 50 terms of a arithmetic series with commo differece of 6 are added together to get a sum of Determie the first ad last terms of this arithmetic sequece. Q7. The sum of the series = How may terms are i this series?
13 MCR3U Uit 8: Sequeces & Series Page 13 of Geometric Series Defiitio: A geometric series is the sum of terms i a geometric sequece. Determiig a Formula for Geometric Series Step 1. Write the sum of terms i a Geometric Series, usig a, r &. S = t 1 + t 2 + t t -2 + t -1 + t Step 2. Multiply the etire expressio by the commo ratio, r. Step 3. You should otice a iterestig patter betwee the equatios i steps 1 & 2. Subtract the equatio i step 1, from the equatio i step 2. Step 4. Simplify ad commo factor each side. Step 5. Rearrage the expressio to fid S.
14 MCR3U Uit 8: Sequeces & Series Page 14 of 14 Formulas for Geometric Series S = ( -1) a r ( r -1) This oly works if r 1. If r = 1, all the terms i the expressio are equal to the first term a, so the S = a 1 1 Q1. Determie the sum of the first 10 terms of the geometric series Q2. Determie the sum of the first 15 terms of a geometric sequece defied by t 3 = Q3. Determie the sum of the geometric series:
15 MCR3U Uit 8: Sequeces & Series Page 15 of 15 Q4. For a geometric sequece, S 1 = 2 ad S 2 = 8. Determie S 5. Q5. The sum of the series = How may terms are i this series? Q6. I a geometric sequece with commo ratio of 4, the sum of the first 7 terms is What is the first term of this sequece?