Fast Five T. Lesso 3 - Arithmetic & Geometric eries & ummatio Notatio Math L - atowski Fid the sum of the first 00 umbers Outlie a way to solve this problem ad the carry out your pla Fid the sum of the first 25 perfect squares Outlie a way to solve this problem ad the carry out your pla 8//2009 Math L - atowski 8//2009 T. - equeces & eries - Lesso 2 (A) Review (B) Arithmetic eries A sequece is a set of ordered terms, possibly related by some patter Oe such patter is called arithmetic because each pair of cosecutive terms has a commo differece The geeral term of a arithmetic sequece is defied by the formula u u + ( - )d A geometric sequece is oe i which the cosecutive terms differ by a commo ratio The geeral term of a geometric sequece is defied by the formula u u x r ( - ) A series is defied as the sum of the terms of a sequece. As a example, start by fidig the sum of the first 00 umbers ad showig a easy way to set it up: 00 2 3 5... 99 00 00 00 99 98 97 9... 2 2 00 0 0 0 0 0... 0 0 o the the sum is (0)(00) 2 8//2009 Math L - atowski 3 8//2009 Math L - atowski.0. - Arithmetic equeces
(B) Arithmetic eries (C) Examples For a arithmetic sequece the the formula for the sum of its terms is: ( a + t) [2 a + ( ) d] 2 2 Ex. Fid the sum of the series3 + 2 + 35 + + 5 Ex 2. For the series 2 + + 20 + 29 +..., fid u 20 ad 20 Ex 3. The fifth term of a arithmetic series is 9 ad the sum of the first is 80. Fid the first three terms of the series. Ex. I a arithmetic series of 50 terms, the 7th term is 53 ad the 28th term is 8. Fid the sum of the series. 8//2009 Math L - atowski 5 8//2009 Math L - atowski (C) Examples (D) ummatio Notatio ex 5. hayla deposits $28 ito her accout. Each week she deposits $7 less tha the previous week util she deposits her last deposit of $2. What total amout did she deposit? ex. Jaye buys 0 widgets o the Ja st, 5 o the st of Feb, 20 o the st of March, etc... How may widgets has she acquired i 2 years? How log does it take her to acquire 5,000 widgets? ummatio otatio is a shorthad way of sayig take the sum of certai terms of a sequece the Greek letter sigma, Σ is used to idicate a summatio I the expressio i represets the term umber (or idex of summatio), ad a i represets the geeral term of the sequece beig summed o therefore, a i a a + a + a + a +... + a i 2 3 8//2009 Math L - atowski 7 8//2009 Math L - atowski 8.0. - Arithmetic equeces
(E) ummatio Notatio Lesso 2 Fast Five Ex Arithmetic eries ( + ) ( + ) + (2 + ) + (3 + ) + ( + ) + (5 + ) + ( + ) ( + ) (2) + (3) + () + (5) + () + (7) ( + ) 27 Determie 7 + 3 + 9 + 27 + 8 + 23 +.. NO CALCULATOR 8//2009 Math L - atowski 9 (E) Examples of ummatio Notatio (E) Examples of ummatio Notatio Write out the series expasio for the followig ad the use the TI-8 to evaluate the sums: 8 0 9 i 2 i i e l ( i) Evaluate the series that are defied i the followig summatio otatios: 20 25 ( 3i 5) + 20 ( 5) 0 25 i 2 ( i 5) + 5 5 ( i) ( i) is 8//2009 Math L - atowski 2.0. - Arithmetic equeces
(F) Geometric eries (F) Geometric eries Fid the sum of the first 7 terms of the series 7 + 3 + 9 + 27 + 8 + 23 + 729 7 a + ar + ar 2 + ar 3 + ar + ar 5 + ar 7 3 3 2 3 3 3 3 5 3 3 7 3 3 2 3 3 3 3 5 3 3 7 3 7-7 2 7 (3 + 3 2 + 3 3 + 3 + 3 5 + 3 + 3 7 ) - ( + 3 + 3 2 + 3 3 + 3 + 3 5 + 3 ) 3 7 - o i geeral, the formula for the sum of a geometric series is: ( u+ u) u ( r ) r r 7 ½(3 7 - ) 8//2009 Math L - atowski 3 8//2009 Math L - atowski (G) Examples (G) Examples ex. Fid 8 give ad rewrite i summatio otatio: (a) 2 - + 8-5 +... (b) 200 + 00 + 50 + 25 +... ex 2. Fid the total amout you make if you were paid a rupee a day, but the amout was doubled every day for a moth ex 3. Fid the sum / + / + + +... + 553 Ex. The fifth term of a geometric series is 05 ad the sixth term is 25. Fid the sum of the first ie terms. ex 5. A ball drops from a height of m ad its height o the bouce is 5/8th of the previous maximum height. Determie the total height bouced by the ball after it touches the groud for the 7 th bouce. 8//2009 Math L - atowski 5 8//2009 Math L - atowski.0. - Arithmetic equeces
(H) Examples of ummatio Notatio (H) Examples of ummatio Notatio Ex 2 Geometric eries Give the series 200 + 00 + 50 + 25 ( ) ()( 0.5) + ()( 0.5) + ()( 0.5) + ()( 0.5) 2 0 2 3 ( ) ()() + ()( 0.5) + ()(0.25) + ()( 0.25) 2 ()( ) + ( 2) + + ( 0.5) 2.5 2 (i) Write a summatio expressio for the series (ii) Determie (iii) Determie 5 (iv) Determie 2 (v) Predict,000,000 8//2009 Math L - atowski 7 8//2009 Math L - atowski 8 (H) Examples of ummatio Notatio (H) Examples of ummatio Notatio The series ½ + ¼ + /8 + / +... is a example of a ifiite geometric series. (a) Determie the sum of this series. (b) Is it possible to fid the sum of ay ifiite geometric sequece? Explai. (c) Uder what coditios is it possible to fid the sum of a ifiite geometric sequece how that the sum of terms of the series 2 + + ½ + ¼ +... u is always less tha, where is ay atural umber. Explai what the followig otatios mea: lim 2 i ( 2 ) 2 i ( 2 ) 2 i ( 2 ) 8//2009 Math L - atowski 9 8//2009 Math L - atowski 20.0. - Arithmetic equeces
(I) Iteret Liks (J) Homework Geometric equeces & eries From West Texas A&M Arithmetic equeces & eries From West Texas A&M U HW: Ex 2E. #ae; Ex 2E.2 #c, 2a, 3,5,, ; Ex 2E.3 #bc, 2cd,,, 7 HW Ex 2F #c, 3c, c, 5ab ad IB packet # - 8 8//2009 Math L - atowski 2 8//2009 Math L - atowski 22.0. - Arithmetic equeces