CHAPTER : ARITHMETIC PROGRESSION CONTENTS. Idetify characteristics of arithmetic progressio PAGE 2.2 Determie whether a give sequece is a arithmetic p
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1 ADDITIONAL MATHEMATICS FORM 5 MODULE ARITHMETIC PROGRESSION
2 CHAPTER : ARITHMETIC PROGRESSION CONTENTS. Idetify characteristics of arithmetic progressio PAGE 2.2 Determie whether a give sequece is a arithmetic progressio 3.3 Determie by usig formula: a) specific terms i arithmetic progressios b) the umber of terms i arithmetic progressios Fid a) The sum of the first terms of arithmetic progressios b) The sum of a specific umber of cosecutive terms of arithmetic progressios c) The value of, give the sum of the first terms of arithmetic progressios. 6.5 Solve problems ivolvig arithmetic progressios 7 SPM QUESTIONS 8-0 ASSESSMENT -2 ANSWERS 3
3 PROGRESSIONS ( Arithmetic Progressio) ARITHMETIC PROGRESSION th The term : T = or T = The sum of the first term: S = or S = a = d = l = = Fill i the blak. Idetify characteristics of arithmetic progressio: EXERCISE : Complete each of the sequece below to form a arithmetic progressio. a) 2,, 8,, 4, 7 b) -2, -5,, -, c) -4,,, 2, 2 d) 2x -3, 2x -,,, 2x + 5 2
4 .2 Determie whether a give sequece is a arithmetic progressio EXERCISE 2 :. Determie whether a give sequece below is a arithmetic progressio. a) 5,, 7, 23,.. ( ) b) -20, -50, -30, -35,. ( ) c), 4, 9, 6, ( ) d) 2x + y, 4x y, 6x -3y, ( ) 2. k +3, 2k + 6,8 are the first three terms of a arithmetic progressio,fid the value of k. Note: If x,y ad z are three terms of a arithmetic progressio, y x = z - y 3. Give that W x 2,5 x,7x 4 are three cosecutive terms of a arithmetic progressio here x has a positive value. Fid the value of x. 4. Give that the first three terms of a arithmetic progressio are 2 y,3y 3 ad 5y+. Fid the value of y. 3
5 .3 Determie by usig formula: a. specific terms i arithmetic progressios b. the umber of terms i arithmetic progressios T a ( ) d EXERCISE 3:. Fid the 9 th term of the arithmetic progressio. 2, 5, 8,.. Solutio: a = 2 d = 5-2=3 2. Fid the th term of the arithmetic progressio. 5 3,,2,... 2 T9 2 (9 )3 = 3. For the arithmetic progressio 0.7, 2., 3.5,..,fid the 5 th term. th 4. Fid the term of the arithmetic progressio 4,6,9, Fid the 7 th term of arithmetic progressio k, 2k +, 3k+2, 4k+3,. 6. Give that arithmetic progressio 9 +6x, 9+4x, 9+2x, Fid the 0 th term. 4
6 EXERCISE 4:. Fid the umber of terms of the arithmetic progressio a) 4,9.4,..,64 2. For the arithmetic progressio 5, 8,,. Which term is equal to 320? T (-)5= =64 5=65 = 3 b) -2, -7, -2,, The sequece 2, 6,,. is a arithmetic progressio. Fid the first egative term i the progressio. c),,,..., d) x y, x y, x 3 y,..., x 3y 4. Fid the umber of terms of the arithmetic progressio a 2 b,3a 3 b,5a 4 b,...,23a 3b 5
7 .4 Fid a) the sum of the first terms of arithmetic progressios. b) the sum of a specific umber of cosecutive terms of arithmetic progressios. c) the value of, give the sum of the first terms of arithmetic progressios. S 2 2 a d o r S = a + l 2 t h l = t h e t e r m EXERCISE 5:. Fid the sum of the first 2 terms of the arithmetic progressio -0, -7, -4, 2. Fid the sum of all the terms of the arithmetic progressio 38, 3, 24,., For the arithmetic progressio -4,, 6,.fid the sum of all the terms from the 6 th term to the 4 th term. How may terms of the arithmetic 2, 8, 3,8, must be take for the sum to be equal to 575? Solutio: Sum of all the terms from the 6 th term to the 4 th = S 4 S 5 = = 6
8 .5 Solve problems ivolvig arithmetic progressios. EXERCISE 6:. The 5 th ad 9 th of arithmetic progressio are 4 ad 6 respectively. Fid the 5 th term. Solutio: T 4 a (5 ) d 4 5 a 4d 4...() T 6 9 a+(9-)d=6 a+8d=6...(2) Solve simultaeous equatio to fid the value of a ad d, the the 5 th term. 2. The 2 d ad 8 th terms of a arithmetic progressio are -9 ad 3 respectively. Fid a) the first term ad the comma differece. b) the sum of 0 terms begiig from the 2 th term. 3. The sum of the first 6 terms of a arithmetic progressio is 39 ad the sum of the ext 6 terms is -69. Fid a) the first term ad the commo differece b) the sum of all the terms from the 5 th term to the 25 th term. 4. The sum of the first terms is give 2 by S 6 3 Fid a) the th term i terms of b) the commo differece 7
9 SPM QUESTIONS:. 2003: Paper : (o. 7) The first three terms of a arithmetic progressio are k 3, k 3, 2k 2 Fid a) the value of k b) the sum of the first 9 terms of the progressio 3 marks : Paper : ( o 0) Give a arithmetic progressio -7, -3,,., state three cosecutive terms i this progressio which sum up to 75 3 marks 8
10 3. Paper : ( o. ) The volume of water i a tak is 450 litres o the first day. Subsequetly, 0 litres of water is added to the tak everyday. Calculate the volume, i litres, of water i the tak at the ed of the 7 th day. 2 marks : Paper : (o. ) The first three terms of a arithmetic progressio are 5, 9, 3 Fid a) the commo differece of the progressio b) the sum of the first 20 terms after the 3 rd term. 4 marks 9
11 5. Paper 2 : ( Sectio A o.3) Diagram shows part of a arragemet of bricks of equal size. Diagram The umber of bricks i the lowest row is 00. For each of the rows, the umber of bricks is 2 less tha i the row below. The height of each bricks is 6 cm. Ali builds a wall by arragig bricks i this way. The umber of bricks i the highest row is 4. Calculate a) the height, i cm, of the wall [ 3 marks ] b) the total price of the bricks used if the price of oe brick is 40 se. [ 3 marks ] 0
12 ASSESSMENT:. The sequece -3,-0,-7,,28,3 is a arithmetic progressio. Fid a) the umber of terms i the progressio b) the sum of all the terms i the progressio. 2. Give that k-, 2k 4 ad k +5 are the first three terms of arithmetic progressio. Fid the commo differece. 3. Calculate how may moths it will take to repay a debt of RM5800 by mothly paymet of RM00 iitially with a icrease of RM20 for each moth after that.
13 4. The 4 th of a arithmetic progressio is 8 ad the sum of the first eight terms of the progressio is 24. Fid a) the first term ad the commo differece b) the sum of all the terms from the 0 th term to the 20 th term. 5. Sum of the first te of the arithmetic progressio is 230 ad sum of te terms after that is 830. Fid the first term ad the commo differece. 2
14 ANSWERS: EXERCISE : a) 5, b) -8,-4 c) 2, 2 d) 2x+,2x+3 EXERCISE 3: T k+6 6. T0 9 2x 4. EXERCISE 5: ASSESSMENT:. a) 49 b) a) 33,-5 b) ,6 EXERCISE 2:. a) Arithmetic progressio b) Not a arithmetic progressio c) Not a arithmetic progressio d) Arithmetic progressio 2. k 3 3. x = 4 4. y = 5 EXERCISE 4:. a) 3 b) 26 c) 22 d) 45 T T EXERCISE 6: a) -,2 b) a) a=4, d= -3 b) a) T=2-3 b) d=2 PAST YEAR QUESTION:. a) k=7 b) S T0 T9 T8 29, 25, 2 3. T a) 4 b) a) 294 b) RM
15 ADDITIONAL MATHEMATICS FORM 5 MODULE 2 PROGRESSIONS (Geometric Progressio) 4
16 CHAPTER 2 : GEOMETRIC PROGRESSIONS Cotets Page 2. CONCEPT MAP (GEOMETRIC PROGRESSIONS) IDENTIFY CHARACTERISTICS OF GEOMETRIC PROGRESSIONS 2.3 DETERMINE WHETHER A GIVEN SEQUENCE IS A GEOMETRIC PROGRESSION 2.4 DETERMINE BY USING FORMULA a) specific terms i geometric progressios b) The umber of terms i geometric progressios 2.5 FIND a) The sum of the first terms of of geometric Progressios b) The sum of a specific umber of cosecutive terms of geometric Progressios c) The value of,give the sum of the first terms of geometric Progressios 2.6 FIND a) The sum to ifity of geometric progressios b) The first term or commo ratio, give the sum to ifiity 2.7 SOLVE PROBLEMS INVOLING GEOMETRIC PROGRESSIONS 6 4 SPM Questios 5 5
17 Assessmet test 8 Aswers 20 CHAPTER 2 : GEOMETRIC PROGRESSION 2. CONCEPT MAP PROGRESSION ARITHMETIC PROGRESSION GEOMETRIC PROGRESSION THE th TERM T T = ar -.a = first term.r = Commo ratio. = umber of terms SUM OF THE FIRST TERMS S = S = a, r > ( r ) r OR a( r r ), r < 6 THE r th TERM T r
18 T r = S r - S r SUM OF INFINITY S = a r, - < r < PROGRESSIONS ( Geometric Progressio) GEOMETRIC PROGRESSION th The term : T = or T = The sum of the first term: S = or S = a = r = l = = Fill i the blak 2.2 Idetify characteristics of geometric progressio: EXERCISE : Complete each of the sequece below whe give a (fist term ) ad r (commo ratio). 7
19 A.r the first four terms of geometric progressio, Example: a) , (-3)(2) = - 6, (-3)(2) 2 = -2, (-3)(2) 3 = -24.b) 3-2.c) 4 3.d) -6-2.e) 2 3.f) 4 y y Determie whether a give sequece is a geometric progressio EXERCISE 2 : 2.3. Determie whether a give sequece below is a geometric progressio. Example :.r = commo ratio a) -8, 4, -2,.r = 4 8 = 2, 2 = 4 2 (true) b) 5,, 7, 23, c) 6, -8, 4,.. d) -20, -50, -30, -35,.. 8
20 e) 2 4 x, x, x f) a 5, a 4 b, a 3 b 2 g),,, h),,, EXERCISE 3 : Give that the first three terms of a geometric progressio are below. Fid the value of x Geometric progressio Value x Example : x 4 2x a).x, x + 4, 2x + 2, = x x 42 (x + 4) 2 = x(2x + 2).x 2 + 8x + 6 = 2x 2 + 2x 0 = x 2 6x 9 0 = (x 8)(x + 2).x 8 = x + 2 = 0 Hece x = x = -2 b) x, x + 2, x + 3 9
21 c) x + 3, 5x - 3, 7x + 3 e) x 6, x, 2x Determie by usig formula: T = ar - EXERCISE 4: 20
22 2.4. specific terms i arithmetic progressios Example :. Fid the 7 th term of the geometric progressio. - 8, 4, -2,.. Solutio: a = - 8 r = T 7 = (-8)( = 8 2 )7-4 8 = 2 2. Fid the 8 th term of the geometric progressio. 6, -8, 4, 3. For the geometric progressio 4 2,,,..,fid the 9 th term Fid the 3 th term of the geometric progressio 50, 40, Fid the 0 th term of geometric progressio a 5, a 4 b, a 3 b 2 6. Give that geometric progressio 5.6,.4, 0.35, Fid the 0 th term. 2
23 2.4.2 Fid the umber of terms of the arithmetic progressio EXERCISE 5: Example : a) 64, - 32, 6,.- 8 b) 2, -4, 8, 52 T = - 8 a = 64, r = = = = = = 9. = 0 3. c) 405, - 35, 45, d) 4 2 8,,..,
24 2.5 Fid d) The sum of the first terms of geometric progressios e) The sum of a specific umber of cosecutive terms of geometric progressios f) The value of, give the sum of the first terms of geometric progressios. S = S = a( r ), r > r OR a( r ), r < r 2.5. Fid the sum of the first terms of geometric progressios EXERCISE 6: geometric progressios Fid the sum of the first term Example :.a) 2, - 4, 8, 4.a = 2 r = 2 = ( ( 2) ) S 7 = ( 2) = 86 b) 5, 0, 20.. S 8 c) 2, -6, 3 S 9 d) 2 4 S x, x, x,
25 2.5.2 The sum of a specific umber of cosecutive terms of geometric progressios EXERCISE 7: geometric progressios Example.a) 4, 2,, 64.a = 4 r = 4 2 = 2 Fid the sum of the first term T = 64 4 = 2 64 = = = 8 = 9 S 9 = 4 / 2 / = , 3, 9,
26 c) 24, 2, 6, The value of, give the sum of the first terms of geometric progressios. EXERCISE 8 : 25
27 2.4 Fid : a) the sum to ifiity of geometric progressios b) the first term or commo ratio, give the sum to ifiity of geometric Example :.a) The first ad 4 th tems of a geometric progressio are 2 Fid the value of r Solutio : T 4 = ad (r 3 ) = (r 3 ) = (r 3 ) = 27 64, Hece r = 3 4 b) The first ad 6 th tems of a geometric progressio are 2 2 ad Fid the value of r c) The commo ratio ad 5 th tems of a geometric progressio are 2 3 Fid the value of a ad
28 progressios. a S r S sum to ifiity a = first term r = commo ratio EXERCISE 9: Fid the sum to ifiity of a give geomertric progressio below:. 24, 3.6, 0.54,. Example: 2 2 6, 2,,, a = 6 2 r 6 3 a S r 6 = = , -27,9,.. 3.,,,
29 EXERCISE 0:. The sum to ifiity of a geometric progressio is 200. Give that the first term is 52. Fid the commo ratio. 2. Give that the commo ratio of geometric progressio is. The sum 25 of the first terms,where is large eough such that r 0 is 75. Fid the first term. Example: The sum to ifiity of a geometric progressio is 8. Give that the first term is 2. Fid a) the commo ratio b) the third term Solutio: a) S 8 a 8 r 2 8 r 3 r= 4 3. The sum to ifiity of a geometric progressio is 600 ad the commo ratio is 0.4. Fid a) the first term b) the miimum umber of terms such that the sum of terms to be more the 599. b) 28
30 T ar 3 =2 4 9 = Express each of the recurrig decimal below as a fractio i its simplest form. Example: = Example: = = =4 99 a) c)
31 b) d) Solve problems ivolvig geometric progressios: EXERCISE : Example: A garderer has a task of cuttig the grass of a law with a area of m. 2 O the first day, he cut a area of 6m. O each successive day, he cuts a area. times the area that he cut the previous day the task is completed. Fid a) the area that is cut o the 0 th day. b) The umber of the days eeded to complete the task. Osma is allowed to sped a allocatio of RM millio where the maximum withdrawal each day must ot exceed twice the amout withdraw the day before. If Osma withdraws RM200 o the first day, determie after how may days the amout of moey allocated will all be used up. Solutio: a) a=6, r=. 30
32 T T 0 ar 6(.)9 = m 2 b) S (. ) > 7.25 log. log 7.25 log. log 7.25 log7.25 > log. >20.79 The smallest iteger value of is 2 hece,the umber of the days eeded to complete the task is 2 days. SPM QUESTIONS: (Paper : No.8) I a geometric progressio, the first term is 64 ad the fourth term is 27. Caculate (a) the commo ratio (b) the sum to ifiity of the geometric progressio. [4 marks] 3
33 (Paper : No.9) Give a geometric progressio 4 y, 2,, p,...,express p i terms of y. y (Paper : No.2 ) Express the recurrig decimal as a fractio i its simplest form. [4 marks] (Paper2: Sectio A: No.6) Diagram 2 shows the arragemet of the first three of a ifiite series of similar triagles. The first triagle has a base of x cm ad a height of y cm. The measuremets of the base ad height of each subsequet triagle are half of the measuremets of its previous oe. y cm xcm Diagram 2 32
34 (a) Show that the areas of the triagles form a geometric progressio ad state the commo ratio. [3 marks] (b) Give that x= 80 cm ad y= 40 cm, 2 (i) determie which triagle has a area of 6 cm, 4 2 (ii) fid the sum to ifiity of the areas, i cm, of the triagles. [5 marks] (Paper : No.0) The first three terms of a sequece are 2, x, 8 Fid the positive value of x so that the sequece is (a) a arithmetic progressio (b) a geometric progressio [4 marks] 33
35 (Paper : No. 2) The sum of the first terms of the geometric progressio 8,24,72,.is Fid (a) the commo ratio of the progressio (b) the value of [4 marks] ASSESSMENT:. The first three terms of a geometric progressio are 2x + 3, x ad x 2 with a commo ratio r, where - < r <. Fid (a) the value of x (b) the sum of the first terms,where is large eough such that r 0 34
36 2. I the progressio 5, 0, 20, 40,. Fid the least umber of terms required such that their sum exceeds The third term ad the sixth term of a geometric progressio are 27 ad 8 respectively. Fid the secod term. 4. I a geometric progressio, the sum of the first five terms is Give that the commo ratio is 2. Fid 3. 8 (a) the first term (b) the sum of all the terms from the fourth to the sixth term. 35
37 5. The third term of a geometric progressio exceeds the secod term by 6 while the fourth term exceeds the third term by 2. Fid the sum of the first 5 terms. ANSWERS: EXERCISE : b) 3, 6, -2, 24 c) 4, 2, 36, 08 d) -6, 2, 24, 48 e) 2, 6, 8, y 4y 6y 64y f),,, EXERCISE 3: b) x = -4 c) x = 3 d) x = x = 8 EXERCISE 5: b) = 9 c) = 8 d) = 7 EXERCISE 7: c) = 8, S 8 = 3280 d) = 9, S 9 = -022 EXERCISE 2: a) true b) false c) true d) false e) true f) true g) false h) true EXERCISE 4:. T 7 = T 8 = T 9 = T 3 = 32 9 b 5. T 0 = 4 a 6. T 0 = EXERCISE 6: b) 275 c) d) 729 EXERCISE 8: b) r = 3 c) a = 3 e) = 6, S 6 =
38 EXERCISE 9: EXERCISE 0: 2. r= a=72 4. a) 360 b) 7 4 a) b) 99 4 c) 33 7 d) 5 99 EXERCISE : 3 days SPM QUESTION:. a) 3 r 4 b) S p 2 y a) r 4 b)i. =5 ii a) x=5 b) x=4 6. a) r=3 b) =7 ASSESSMENT:. a) x=3 b) a) -2 b)
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