MST : Supplemetary resource material for Chapter A Sequeces For each of the followig sequeces (a) 9 (b) (c) 8 7 Write ow the et terms of the sequece Write ow a epressio for the close form of the sequece takig the rage variable as If you have a graphical calculator use the seq comma to check that your epressio for the close form correctly geerates the sequece Use your epressio for the close form to preict the th term What special ame is give to the terms of each sequece? For each of the followig arithmetic sequeces (a) 9 7 (b) 9 7 (c) () 8 (e) 8 If you have a graphical calculator evise a meas of geeratig each of the above sequeces by repeately pressig the ENTER key Write ow the et terms of each sequece Defie the arithmetic sequece as a recurrece system with startig value u Write ow a epressio for the close form of the sequece If you have a graphical calculator use the seq comma to check that your epressio for the close form correctly geerates the sequece Use your epressio for the close form to preict the th term of each sequece For each of the followig arithmetic sequeces (a) 7 9 (b) 9 (c) 79 7 8 () (e) 7 8 If you have a graphical calculator evise a meas of geeratig each of the above sequeces by repeately pressig the ENTER key Write ow the et terms of each sequece Defie the arithmetic sequece as a recurrece system with startig value u Write ow a epressio for the close form of the sequece If you have a graphical calculator use the seq comma to check that your epressio for the close form correctly geerates the sequece Use your epressio for the close form to preict the th term of each sequece
For each of the followig geometric sequeces (a) 8 (b) 9 7 8 (c) 8 () (e) y y y y If you have a graphical calculator evise a meas of geeratig each of the sequeces (a) () by repeately pressig the ENTER key Write ow the et terms of each sequece Defie the geometric sequece as a recurrece system with startig value u a rage variable Write ow a epressio for the close form of each sequece If you have a graphical calculator use the seq comma to check that your epressio for the close form correctly geerates the sequece Use the close form to etermie the th term of each sequece Fi the sum of the first fiftee terms of each sequece Describe the log-term behaviour of each of the sequeces For each of the followig geometric sequeces (a) 8 (b) 8 9 78 (c) 8 () 8 (e) y y y y If you have a graphical calculator evise a meas of geeratig each of the sequeces (a)-() by repeately pressig the ENTER key Write ow the et terms of the sequece Defie the geometric sequece as a recurrece system with startig value u Write ow a epressio for the close form of the sequece with If you have a graphical calculator use the seq comma to check that your epressio for the close form correctly geerates the sequece Use your epressio for the close form to preict the th term of the sequece Fi the sum of the first fiftee terms of each sequece Describe the log-term behaviour of each of the sequeces The geeral liear recurrece a r ( a r ) has a close form s r t where s a a t Use this result to r r write ow a epressio for the close form of each of the sequeces give below usig as the rage variable I each case use the give recurrece system to geerate the first terms of the sequece a check that your close form prouces the same sequece Use the close form to evaluate (a) (b) (c)
() 7 8 7 9 (e) 7 The geeral liear recurrece a r ( a r ) has a close form s r t where s a a t Use this result to r r write ow a epressio for the close form of each of the sequeces give below usig as the rage variable I each case use the give recurrece system to geerate the first terms of the sequece a check that your close form prouces the same sequece Use the close form to evaluate (a) 7 9 (b) (c) 9 () 79 (e) 8 Takig the rage variable as fi a close form for each of the followig liear recurreces: (a) 7 (b) 7 9 (c) 9 7 9 () Miscellaeous Problems: 9 The first three terms of a geometric progressio are respectively the first ith a eleveth terms of a arithmetic progressio Give that the terms of the geometric progressio are all ifferet fi the value of the commo ratio r Give that ( ) ( ) a ( 9) are cosecutive terms of a geometric sequece etermie the value of the commo ratio of the sequece A sequece of umbers { u } is efie for by the recurrece relatio u ku where k is a costat Give that u (a) Fi epressios i terms of k for u a u (b) Give also that u fi the possible values of k (c) Give also that the sequece oscillates calculate the value of u A sequece of umbers t t t t is forme by takig a startig value of t a usig the rule t t for k k k (a) If t calculate t t a t Show that t a write ow the value of t (b) If t show that all the terms of the sequece are the same a fi the other value of t for which all the terms are the same (c) Discuss the log term behaviour of the sequece i the cases whe (i) t (ii) t (iii) t
Aswers (a) 9 8 u seq ( ) u square umbers (b) 8 ( u ) seq ( ( ) ) u triagular umbers (c) 79 u seq ( ) 9 u cubes (a) ENTER ENTER ENTER ENTER 9 7 u u u u seq ( ) u (b) ENTER 8 ENTER ENTER ENTER 9 7 7 8 u u 8 u u 8 seq ( 8 ) u 8 (c) ENTER ENTER ENTER ENTER u u u u seq ( ) u 99 () (-) ENTER ENTER ENTER ENTER 8 u u u u seq ( ) u 98 (e) ENTER ENTER ENTER ENTER 7 8 9 7 9 u u u u 9 seq ( 9 ) u 98 (a) 7 ENTER ENTER ENTER ENTER 7 9 u u u 7 u 7 seq ( 7 9) th term u (b) ENTER 7 ENTER ENTER ENTER 7 8 7 u u 7 u u 7 seq ( 7 9) th term u 7 (c) 79 ENTER 7 ENTER ENTER ENTER 7 u u 7 u 79 u 79 7 seq ( 79 7 9) th term u 9 () ENTER ENTER ENTER ENTER 8 u u u u seq ( 9) th term u 98 (e) ENTER ENTER ENTER ENTER 9 7 8 9 u u u u seq ( 9) th term u 99 (a) ENTER ENTER ENTER ENTER 8 u u u u
seq ( ) th term 8 s 77 Diverges (b) ENTER ENTER ENTER ENTER 79 87 98 99 u u u u seq ( ) th term 897 s 9 Diverges (c) ENTER ENTER ENTER ENTER 8 u u u seq ( ) u th term 8 77 s 8 Coverges to () ENTER ENTER ENTER ENTER 9 777 799 u u u u seq ( ) th term 797 9999 s 9 Coverges to 797 (e) [TI-89 users oly] ENTER y ENTER ENTER ENTER 7 8 9 y y y y y u yu u u y z [TI-89 users oly] seq ( y z ) th ( y ) term y s y Diverges if y > ; Costat sequece for y ; Coverges to if < y < ; Costat sequece for y ; Oscillates betwee positive a egative values but coverges to if < y < ; Forms a two cycle betwee ± if y ; Oscillates betwee positive a egative values a iverges for y < (a) ENTER ENTER ENTER ENTER 8 u u u u seq ( 9) th term u 89 s 8 Diverges (b) ENTER ENTER ENTER ENTER 7 88 9 98 78 u u u u seq ( 9) th term u 88 s 778 Diverges (c) ENTER ENTER ENTER ENTER 8 u u u u ( ) seq ( ( ) 9) th 98 term u 89 s 89 Coverges to () ENTER ENTER ENTER ENTER 78 u u u u seq ( 9) th term u s 778 Diverges (e) [TI-89 users oly] ENTER y ^ ENTER ENTER ENTER 8 y y y y y u y u u z u y [TI-89 users oly] seq ( y z 9) th 8 ( y ) term u y s Diverges if y > a coverges y
to if y < ; costat sequece for y (a) ; ENTER As ENTER ENTER ENTER ; 7 9 ; seq ( ) ; 9788 (b) ; ENTER As ENTER ENTER ENTER ; 7 9 ; seq ( ) ; 7 (c) ; ENTER As ENTER ENTER ENTER ; 9 79 9 79 ; seq ( ) ; 9788879 () 7 ; 7 ENTER 7As 8 ENTER ENTER ENTER; 7 99 7 97 ; seq ( 7 ) ; 99987 (e) 9 ; (-) ENTER 9As ENTER ENTER ; 979 ; seq ( 9 ) ; 98879 7 (a) (7 ) ; 7 ENTER As ENTER ENTER ENTER 7 9 8 9 ; seq ( (7 ) ) ; 877 8 87 (b) ; ENTER As ENTER ENTER ENTER; 7 ; seq ( ) ; 88897 (c) 8 ; 9 ENTER As ENTER ENTER ENTER; 9 7 7 9 ; seq (8 ) ; 998977 () 7 ( ) ; 79 ENTER ( ) As ENTER ENTER; 79 9 79 9 ; seq (7 ( ) ) 8 79 (e) (8 9 7) ; ENTER 9As - ENTER ENTER; 9 7 9 9 ; seq ( (8 9 7) ) ; 9 8799889 87 8 (a) 7 (b) (c) ( 7 ) () 9 9 Elimiate from a 8 ar a r to give r ar to give r r a Reject Solve 9 to give The r (a) u k u k k (b) (c) Takig k gives u 8 (a) t t t t t (b) With t t Hece all terms will be Put tk tk t to give t t with roots a Hece the other value of t is
(c) (i) The sequece iverges (ii) t for i i (iii) The sequece gives a -cycle with values ( ± )