ubject: eries ad equeces 1: Arithmetic otal Mars: 8 X - MAH Grade 1 Date: 010/0/ 1. FALE 10 Explaatio: his series is arithmetic as d 1 ad d 15 1 he sum of a arithmetic series is give by [ a ( ] a represets the first term of this series: a represets the umber of terms i this series: 10? d represets the costat differece betwee the terms of this series: d he [ () ( 1)() ] [ ] [ 0 ] he sum of this series must be larger tha 00. herefore [ 0 ] > 00 [ 10 ] > 00 10 > 00 10 00 > 0 ( 10)( 0) > 0 As the umber of terms caot be egative or a fractio, > 10, i.e. oly eleve or more terms will give a sum of larger tha 00.. FALE Explaatio: he first ad secod terms of a arithmetic progressio are 1 ad 1. herefore d 1 1 1 he th term of a arithmetic progressio: a ( a is the first term: a 1 d is the costat differece: d is the term umber: the 5 th term must be foud, therefore 5 herefore Page 1
1 (5 1)() 5 1 ( ) 1 0 17 he thirty-fifth term is therefore 17 ad ot 1.. B Explaatio: First determie if this is a arithmetic or a geometric series: d d 1 1 5 5 r 1 r 1 1 here is a costat differece betwee successive terms ad ot a costat ratio, therefore this series is arithmetic. he sum formula for a arithmetic progressio: a ( a represets the first term of this series: a l represets the last term of this series: l 5 [ ] OR [ a l ] represets the umber of terms i this series:? d represets the costat differece betwee the terms of this series: d 5 represets the last term of this series: 5 is the sum of terms of this series:? he umber of terms must be determied before the sum ca be calculated. he th term of aarithmetic progressio: a ( 5 ( 1)(5) 5 5 5 5 55 Now [ () ( 1)(5) ] [ 8 50 ] 58 5 OR [ ] 58 1 1. C 8 Explaatio: he multiples are: ; 7 ; 0 ; ; 7 ; 10 his is a arithmetic progressio as is added to each term to get the ext term. o fid the umber of terms to be added, you eed to use the -formula of a AP: Page
a ( 10 ( 1)() 10 Now ( 1) [ a d ] () ( 1)() 8 [ ] [ ] 7 7 OR [ l ] [ 10 ] a 7 7 5. D Explaatio: o determie whether a sequece is arithmetic, you eed to fid the differece (d) betwee cosecutive terms. A: d 1 or d or d 7 5 7 5 he aswers are the same, therefore sequece A is arithmetic. B: d 1 or d 7 18 he aswers are ot the same, therefore sequece B is ot arithmetic. C: d 1 or d (x ) ( x ) (5x ) (x ) x x 5x x x 1 x 1 he aswers are the same, therefore sequece C is arithmetic.. 1 8 Explaatio: 1 ( 1) [(1) 1] [() 1] [() 1]... 1 5... he terms of this series differ with, this series is therefore arithmetic. he sum formula for a arithmetic progressio: Page
a represets the first term of this series: a 1 l represets the last term of this series: l - 1 [ a ( ] OR [ a l ] represets the umber of terms i this series:? d represets the costat differece betwee the terms of this series: d represets the last term of this series: - 1 is the sum of terms of this series: Now [ a ( ] [ (1) ( 1)() ] [ ] OR [ a l ] [ 1 ( 1) ] [ ] ± 1 ± 1 As represets the umber of terms, - 1 is a ivalid solutio. 7. - Explaatio: he -formula for a arithmetic progressio: a ( a is the first term: a d is the costat differece: d -5 is the umber of terms to be added: 1 herefore 1 () (1 1)( 5) 1 1 (18)( 5) 1 0 1 ( 8) 1 ( ) 8. 0,5 Explaatio: Arithmetic Mea 1 1 5 1 0, 5 Page
. (1) 1,,, 7,, 18 () is ot () Lucas Explaatio: 1. his is a secod order differece equatio, where the first two terms are give. 1, 1 1 1 1 1 1 7 5 7 5 7 18. he differece betwee the terms is ot costat. herefor this is ot a arithmetic patter.. Eduoardo Lucas discovered this umber patter, whe he compared differet startig poits to the better ow Fiboacci series. How ice to have a umber patter amed after you! 10. (1) 7, 10 () quadratic () - 1, 1 - Explaatio: 1. tudy the patter: -,, 1, 0, 50... Now determie the differeces: 8 1 1 0 Determie the differeces of the differeces: o determie the ext two terms, this patter of differeces must be maitaied. Hece the sixth ad seveth terms must be ad 8 larger tha its predecessors respectively. 50 7 ad 7 7 8 10. A liear patter has its first differece costat. A quadratic patter has its secod differece costat. A cubic patter has its third differece costat.. he formula - 10 geerates the patter, but it is a explicit formula, ie ot a recursive formula. he formula 1, 1 - geerates 1-1 () - 8 () 1 1 () 1 1 0 5 (5) 0 0 50 his is the patter required.. he formula 1 1, 1 - geerates: 1-1 1 (-) 1-1 1 1 () 1 1 18 his is ot the patter required. Page 5
. (1) a 1d 1 () 5 () - Explaatio: ubstitute the two sets of give iformatio ito the -formula for a arithmetic progressio: a ( th term is : a ( 1 a 5d... [1] th 1 term is 1 : a (1 1 1 a 1d... [] ubtract [1] from []: 1 a 5d 1 a 1d... []... [1] 5 7d 5 d a represets the first term of the sequece. o fid the value of a, substitute d 5 ito -1 a 5d: 1 a 5( 5) 1 a 5 a a 1 5 1. - 10-10 Explaatio: he th term of a arithmetic sequece: a ( a is the first term of the sequece, therefore a -7. d is the differece betwee the terms of the sequece: d or 1 ( ) ( 7) 7 d ( 1) ( ) 1 herefore: 7 ( 1)() [ remais as it is uow ] 7 10 1. 10 Explaatio: t 1 (t ) [(1) ] [() ] [() ]... [( ) ] 1 1... ( ) Page
his is a arithmetic series with d. he sum formula for a AP: [ a ( ] a represets the first term of this series: a 1 represets the umber of terms i this series: d represets the costat differece betwee the terms of this series: d is the sum of terms of this series: < 58 Now [ ( 1) ( 1)() ] [ ] [ ] herefore < 58 58 < 0 ( )( ) < 0 As represets the term umber of the last term, oly 1; ; ; ; ca be valid values for ( < < ). herefore, the greatest value that could have is. 1. A Explaatio: herefore : ( 8) 8 15. B 5 Explaatio: I this arithmetic sequece: a, d?, ad he... a ( ( 0 d 0 d herefore: Page 7
x ad y 1 d 0 d 0 15 Questios, 8 Pages Page 8