Auchmuty High School Mathematics Department Sequences & Series Notes Teacher Version

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1 equeces ad eies Auchmuty High chool Mathematics Depatmet equeces & eies Notes Teache Vesio A sequece takes the fom,,7,0,, while 7 0 is a seies. Thee ae two types of sequece/seies aithmetic ad geometic. Aithmetic A aithmetic sequece is oe whee we establish the ext umbe i the sequece by addig (o subtactig) a give umbe. This umbe is kow as the commo diffeece. Geeally: Let a be the fist tem i the sequece ad d be the commo diffeece. u a u a d u u d a d u u d a d u a d th This fomula is impotat as it allows us to fid the tem i a aithmetic sequece povided we kow the fist tem ad the commo diffeece. If we sum the tems of this sequece we have a aithmetic seies. The sum to tems of this aithmetic seies is give by the fomula Poof u u u u u Let Rewitig this backwads u u u u u Addig these togethe u u u u u u u u u u a a d a d a d a d a d a d a d a d a a d a d a d a d a d a d a d

2 Auchmuty High chool Mathematics Depatmet equeces & eies Notes Teache Vesio Examples (a) Fid the 0th tem of the aithmetic sequece,,8,,, (b) Fid the sum of the fist eight tems. u a d (b) (a) a, d 7 a d u u (a) Fid the 0th tem of the aithmetic sequece (b) Fid the sum of the fist fiftee tems.,,,, (a) a, d u a d u 0 0 a d 7 (b) u (a) Fid the 7th tem of the aithmetic sequece 0,,0,, (b) Fid the sum of the fist 00 tems of the seies. u a d (b) (a) a 0, d a d 00 u u (a) A sequece is give as follows : x,x,x 7,x 0, tate a expessio fo the commo diffeece. (b) Fid a expessio fo the th tem. (c) Fid a expessio fo the fist tems of the seies.

3 (a) Commo diffeece x (b) u a d u x x x (c) x x Auchmuty High chool Mathematics Depatmet equeces & eies Notes Teache Vesio 8 x 7 0x 9 If the 6th tem of a aithmetic sequece is ad the d is 0, defie the sequece ad fid the fist fou tems. u6 a d u a d a d 0 a d olve equatios simultaeously to get a, d u a d u u, u 6, u 0, u 6 Fid the value of whe a, d ad the tem of the sequece is. u a d Which tem i the aithmetic sequece,,, is -? u a d u 7 The 7th tem is -.

4 Auchmuty High chool Mathematics Depatmet equeces & eies Notes Teache Vesio 8 If a aithmetic sequece has u ad u, which tem has the value? u a d 60 6 The 6st tem has the value. 9 A aithmetic sequece stats with. The 00th tem is 9. What is the commo diffeece betwee the tems? u a d u d u00 99d 9 99d 99d 96 d The commo diffeece is. 0 The sum of the fist tems of a aithmetic sequece is 6. The sum of the fist tems is. What is the sum of the fist 0 tems? a d : 6 a d : a d 6 a 6d a 66d olve simultaeously to get a, d

5 Auchmuty High chool Mathematics Depatmet equeces & eies Notes Teache Vesio The fist two tems of a aithmetic sequece ae 9 ad i that ode. (a) Fid the sum of the fist (i) 8 tems (ii) 9 tems (b) Hece calculate the 9th tem. (a) a 9, a d d (b) 9th tem The fist two tems of a aithmetic sequece ae ad 0. Afte what tem does this seies exceed 00? a, 8 00 a d d Whe is o 8 eies exceeds 00 at the 9th tem. The sum of the fist 0 tems of a aithmetic sequece is 000. The commo diffeece is. What is the fist tem? 0 0 a a 98 a 7 The fist tem is 7.

6 Auchmuty High chool Mathematics Depatmet equeces & eies Notes Teache Vesio Questios (a) Fid the th tem of the sequece,6,0,, (b) Fid the sum of the fist tems. The d tem of a aithmetic sequece is 0 ad the th tem is 0. (a) Fid the fist tem ad commo diffeece. (b) Fid the 0 th tem of the sequece. The th tem of a aithmetic sequece is ad the 7 th tem is 7. (a) Fid a fomula fo the th tem of the sequece. (b) What is the sum of the fist 0 tems? If a aithmetic sequece has u 8 ad u, which tem i the sequece has the value 9? The st tem i a aithmetic sequece is 6. The 0 th tem is. What is the commo diffeece betwee the tems? 6 I the aithmetic sequece begiig,8,,0, exceed 00?, which tem is the fist to 7 The st tem of a aithmetic sequece is ad the 6 th tem is twice the d tem. Fid the commo diffeece ad the th tem. 8 Fo the aithmetic seies 7 9 (a) Fid a fomula fo, the sum of the fist tems. (b) Hece, fid how may tems must be take to give a sum of 9. 9 (a) Fid a fomula fo, the sum of the fist tems of the aithmetic seies 88 (b) Hece fid the least umbe of tems that ae equied to make a sum exceedig I a aithmetic sequece, the 8 th tem is twice the th tem. (a) If a is the fist tem ad d is the commo diffeece, show that a d. (b) Give that the 0th tem is 0, fid the sum of the fist 0 tems of this sequece. 6

7 Auchmuty High chool Mathematics Depatmet equeces & eies Notes Teache Vesio Geometic A geometic sequece is oe whee we establish the ext umbe i the sequece by multiplyig o dividig by a give umbe. This umbe is called the commo atio. It is the atio betwee two cosecutive tems. Geeally : Let a be the fist tem of the sequece ad be the commo atio. u u u a a a u a th Agai, this esult should be emembeed. It allows us to fid the tem i a geometic sequece povided we kow the fist tem ad the commo atio. If we sum the tems of this sequece we have a geometic seies. The sum to tems of this geometic seies is give by the fomula Poof a a a a a a a, If we multiply both sides by we get a a a a a ubtactig a a a a, (Whe, the seies is aaaa so a ) Examples Fid the th tem ad the 0th tem of the geometic sequece,,8, a,, u u a u

8 Auchmuty High chool Mathematics Depatmet equeces & eies Notes Teache Vesio Fid the geometic sequece whose d tem is 8 ad whose 8th tem is 7. u 8 u a 7 a 7 a 7 a 8 8 a a o u u, u 6, u 8, u, o sequece is,6,8,, Give the geometic sequece,0,0,0, fid the value of fo which u 080. a, 0, u a u l l 096 l l 096 l 096 l Fo each of the followig geometic sequeces (i) idetify a ad (ii) fid a expessio fo the th tem. (a) 8,86,6,, (b), 8,6,, (c) 06,0 8,0 0, (d),, 8, 6, 7 7 (a) (i) a 8, 86 (ii) 8 u u a 8 8

9 Auchmuty High chool Mathematics Depatmet equeces & eies Notes Teache Vesio (b) (i) a, 8 (ii) u u a (c) (i) a 06, (ii) u u a (d) (i) a, (ii) u u a (a) The fist tem of a geometic sequece is. The teth tem is 9. Calculate the commo atio to d.p. (b) The commo atio of a geometic sequece is The 8th tem is. What is the fist tem to the eaest whole umbe? (a) a, u0 9 (b) 0 98, u8 u u8 a0 98 a0 98 a Fid the sum of the fist 0 tems of the geometic seies a,

10 Auchmuty High chool Mathematics Depatmet equeces & eies Notes Teache Vesio 7 Fid the sum to six tems of the geometic sequece whose fist tem is ad whose commo atio is. a, A geometic sequece stats,,8 7, is bigge tha 00?. What is the smallest value of fo which a, l l l 7 l 06 6 as 9 A geometic seies is such that ad 6 6. Idetify the seies. a ad a so o so The seies is 8. The th tem is give by u. 0

11 Auchmuty High chool Mathematics Depatmet equeces & eies Notes Teache Vesio 0 Calculate each of these geometic seies to the equied umbe of tems. (a) 0 to 6 tems. (b) 0 to 7 tems. (c) to 8 tems. (a) a, (b) a, (c) a 0, A pedulum is slowly comig to est. With each swig the ac it sweeps is educed by %. Each swig takes the same time, secods. The fist sweep is a ac. (a) How may degees has the pedulum swug though i five swigs? (b) How log will it take befoe a total exceedig 90 has bee swept though? (a) The sequece is,76, a, (b) l098 l0 8 l 08 (l098 is ve so evese iequality sig) l whole swigs ae eeded befoe a total exceedig 90 has bee swept though. This will take 9 08 secods.

12 Auchmuty High chool Mathematics Depatmet equeces & eies Notes Teache Vesio Questios (a) Fid a expessio fo the th tem of the geometic sequece, 8,6,, (b) Calculate the 0th tem. A geometic sequece of positive tems has d tem 8 ad 7 th tem 8. Fid the th tem of this sequece. Give the geometic sequece 6,,,8,, fid the value of fo which u 980. A geometic sequece of positive tems has st tem 9 ad d tem 6. Fid the 0 th tem of this sequece. Fid the sum to 9 tems of the seies Evaluate the sum of the geometic seies A geometic sequece has d tem ad 6 th tem 8. Fid the sum of the fist 8 tems of this sequece. 8 A geometic sequece has th tem 8 ad 8 th tem. Fid the sum of the fist 6 tems of this sequece. 9 (a) Fid a expessio fo the sum of the fist tems of the geometic sequece 0,0,90,70, (b) Fid the umbe of tems which must be added to give a sum of 800. (c) Fid the least umbe of tems which must be added to give a sum exceedig Fid the least umbe of tems of the geometic seies 86 which must be added to give a sum exceedig A geometic sequece has st tem ad th tem 0. Fid a expessio fo the sum to tems of this sequece ad hece, fid the least umbe of tems which must be added to give a sum exceedig A lie cm log is divided ito 6 pats such that the legths of the pats fom a geometic sequece. Give that the legth of the logest pat is times the legth of the shotest pat, fid the legth of the shotest pat. Fo a cetai sequece it has bee foud that the sum to tems is Fid the type of sequece a the value of u 6, the 6 th tem.,.

13 Auchmuty High chool Mathematics Depatmet equeces & eies Notes Teache Vesio Covegece ad Divegece of Ifiite Geometic eies Examples Coside the geometic seies If we wee to cotiue we would see that 0 as. This is a coveget seies. We wite I this case as. We wite This time as, cotiues to icease. This is a diveget seies. does ot exist.

14 Auchmuty High chool Mathematics Depatmet equeces & eies Notes Teache Vesio Coditio fo Existece of um to Ifiity Coside the fomula fo. a, Whe, 0 as. o as, 0 a a a The sum to ifiity of a geometic seies exists if ad ca be calculated by a Examples Fid the sum to ifiity of the seies 6 8 a, 6 so exists. a 6 Fid the sum to ifiity of the seies 6 8 a, 6 so exists. a 6 Questios Explai why the geometic seies 7 8 has a sum to ifiity ad fid the sum to ifiity. A geometic sequece has st tem 8 ad a sum to ifiity of 6. Fid the th tem of the sequece. A geometic seies has a commo atio of ad the sum to ifiity of the seies is. Fid the st ad the th tems of this seies.

15 Auchmuty High chool Mathematics Depatmet equeces & eies Notes Teache Vesio igma Notatio The otatio b f k is shothad fo f a f a f a f b ka b ae iteges ad a Examples b. whee a ad k k k k Fid k k 7 It is best to wite out the fist few tems to see what kid of seies it is ad get the fist tem ad commo diffeece o atio. The seies is 7 summed to 0 tems. It is aithmetic with a, d ad 0. a d so 0 6 Fid The seies is It is geometic with a,. so exists. a so

16 ome Useful Popeties Auchmuty High chool Mathematics Depatmet equeces & eies Notes Teache Vesio f k g k f k g k k k k af k a f k, whee a is a costat. k k ome Useful tadad Fomulae k k k k tems k k k 6 ak b ak b k k k a b No eed to memoise these as they ae i the fomula list! Notice that k k k k Examples 0 Evaluate k k k k k k k k k k usig fomulae fom above. 6

17 Auchmuty High chool Mathematics Depatmet equeces & eies Notes Teache Vesio Evaluate the sum of the seies k k k k k k k k 6 9 k 6k 9k k 6 k 9 k k k k Obtai a expessio fo k. k Give you aswe as a sigle algebaic factio i its simplest fom. k k k k k 6 7 Evaluate 0 k k 0 0 k k k k k k

18 Auchmuty High chool Mathematics Depatmet equeces & eies Notes Teache Vesio Questios Evaluate (a) k k (b) k k k 8 0 (c) k k k k k (d) k k 8 k Evaluate the sum of these seies by witig them i the fom (a) (b) (c) 6 0 k f k Fid a expessio fo these summatios i simplest fom. (a) k (b) kk (c) kk (d) k k k k k k k Evaluate (a) 60 k 6 k 0 (b) k (c) k k k 6 Diffeetiatio ad Itegatio of eies Examples Fid the sum to ifiity of x x x x, x Hece, use itegatio to fid a seies fo l x. x x x x is geometic with a ad x, x so a sum to ifiity exists. x x x x x o Itegatig both sides. Put x 0 to get 0 x C so x x dx x x x x dx x x x l x x C x x x l x x, whe x 8

19 Auchmuty High chool Mathematics Depatmet equeces & eies Notes Teache Vesio 99 Fid the sum of e x e x e x e x, x 0 Use diffeetiatio to fid a fomula fo the seies The seies is geometic with a,. x e ad 00 e e e e x x 99 x x 99 x, 0 a so 00 00x e 00x e, x 0 x x e e 00x x x x 99x e o e e e e, x 0 x e Diffeetiatig each side 00e e e e x x x 99x e e e 99e x e 00x x 00 x x x x x 99x 99e 00e e e e e 99e x e 0x 00x x 9

20 Past Pape Questios Auchmuty High chool Mathematics Depatmet equeces & eies Notes Teache Vesio 00 A0 Defie x by whee is a positive itege. Expess i tems of. By cosideig x x x x x x x, show that x x x x, x. Obtai the value of lim. (,, maks) 00 A Give that u k, k k, obtai a fomula fo u. Fid the values of fo which. (, maks) 00 Q6 (a) Obtai the sum of the seies 8 6. (b) A geometic sequece of positive tems has fist tem, ad the sum of the fist thee tems is 66, Calculate the commo atio. (c) A aithmetic sequece, A, has fist tem a ad commo diffeece, ad a geometic sequece, B, has fist tem a ad commo atio. The fist fou tems of each sequece have the same sum. Obtai the value of a. Obtai the smallest value of such that the sum to tems fo sequece B is moe tha twice the sum to tems fo the sequece A. (,,, maks) k k 00 Q The sum,, of the fist tems of a sequece, u, u, u, is give by 8,. Calculate the values of u, u, u ad state what type of sequece it is. Obtai a fomula fo u i tems of, simplifyig you aswe. (, maks) 0

21 006 Q6 Auchmuty High chool Mathematics Depatmet equeces & eies Notes Teache Vesio The fist thee tems of a geometic sequece ae x x x x, x x ad x x x, whee x. (a) Obtai expessios fo the commo atio ad the th tem of the sequece. (b) Fid a expessio fo the sum of the fist tems of the sequece. (c) Obtai the age of values of x fo which the sequece has a sum to ifiity ad fid a expessio fo the sum to ifiity. (,, maks) 007 Q9 6. how that q. Hece wite dow a fomula fo 6 how that q 6 q 9q. (,, maks) q 008 Q The fist tem of a aithmetic sequece is ad the 0th tem is 97. Obtai the sum of the fist 0 tems. ( maks) 009 Q The fist two tems of a geometic sequece ae a p ad a p. Obtai expessios fo ad i tems of p, whee k k a. j j Give that 6 show that p 6. Give also that a p ad that p 0, obtai the exact value of p ad hece the value of. (,,,, maks)

22 Auchmuty High chool Mathematics Depatmet equeces & eies Notes Teache Vesio 00 Q The secod ad thid tems of a geometic seies ae 6 ad espectively. Explai why the seies has a sum to ifiity ad obtai this sum. ( maks) 0 Q8 Wite dow a expessio fo ad a expessio fo 0 Q The fist thee tems of a aithmetic sequece ae Obtai the value of a ad the commo diffeece.. (, maks) a,, whee a 0. a Obtai the smallest value of fo which the sum of the fist tems is geate tha 000. (, maks) 0 Q The fist ad fouth tems of a geometic sequece ae 08 ad 6 espectively. Calculate the value of the commo atio. Give that the sum of the fist tems is 088, fid the value of. 0 Q7 Wite dow the sums to ifiity of the geometic seies x x x ad x x x valid fo x. Assumig that it is pemitted to itegate a ifiite seies tem by tem, show that, fo x, x x x l x x how how this seies ca be used to evaluate l. Hece detemie the value of l coect to decimal places. (, maks) (7, maks)

23 Auchmuty High chool Mathematics Depatmet equeces & eies Notes Teache Vesio 0 Q The sum of the fist twety tems of a aithmetic sequece is 0. The twety-fist tem is 7. What is the sum of the fist te tems? ( maks) 06 Q A geometic sequece has secod ad fifth tems 08 ad espectively. (a) Calculate the value of the commo atio. (b) tate why the associated geometic seies has a sum to ifiity. (c) Fid the value of this sum to ifiity. (,, maks) 07 Q The fifth tem of a aithmetic sequece is -6 ad the twelfth tem is -. (a) Detemie the values of the fist tem ad the commo diffeece. (b) Obtai algebaically the value of fo which = -. (, maks)