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1 addtioalmathematicsprogressiosad dtioalmathematicsprogressiosaddti oalmathematicsprogressiosaddtio almathematicsprogressiosaddtioal PROGRESSION mathematicsprogressiosaddtioalma Name thematicsprogressiosaddtioalmathe... maticsprogressiosaddtioalmathema ticsprogressiosaddtioalmathematic sprogressiosaddtioalmathematicspr ogressiosaddtioalmathematicsprogr essiosaddtioalmathematicsprogress iosaddtioalmathematicsprogressio saddtioalmathematicsprogressiosad dtioalmathematicsprogressiosaddti oalmathematicsprogressiosaddtio almathematicsprogressiosaddtioal mathematicsprogressiosaddtioalma thematicsprogressiosaddtioalmathe
2 PROGRESSION. ARITHMETIC PROGRESSION. Idetify characteristics of arithmetic progressio ( AP) AP is a sequece of umbers where the differece betwee successive terms is a costat. Example : examples of AP 5, 7, 9,, ,, 7, 3, Determie whether a give sequece is a AP If a sequece is a AP, the the differece betwee successive terms is a costat. Example : Determie whether each of the followig sequeces is a AP a., 7, 0, 3, 6, It is a AP b.,, 9, It is ot a AP Exercise : Determie whether each of the followig sequeces is a AP a. 3, 5, 7, 9,.. b.,,,,... 8 c. 0, 7,,,.. d. a, a + b, a + b, a + 3b,.. Homework : From Text Book- page 0, Exercise., o ad. zefry@sas.edu.my
3 .3 Determie by usig formula a. specific terms i AP b. the umber of terms i AP A AP ca be represeted as a, a + d, a + d, a + 3d, a + d, First term, a Secod term, T Third term, I geeral, the th term, T is represeted by T 3 Fourth term, T Fifth term, T 5 T = a + ( ) d where a first term d commo differece umber of terms Example 3 : a. Fid the 8 th term of the AP, 7, 0, 3, 6,. a =, d = 3 T = a + ( ) d T 8 = + ( 8-) 3 = 5 b. Fid the th term of the AP, 7, 3, -, a =, d = - T = a + ( ) d T = + ( -)(-) = - 33 c. Fid the th term of the followig AP 3, 8, 3, 8, a = 3, d = 5 T = a + ( ) d = 3 + ( -) 5 = = 5 d. Fid the umber of terms i the AP 5, 8,,, a = 5, d = 3, T = T = a + ( ) d = 5 + ( -) 3 = = 0 zefry@sas.edu.my 3
4 Exercise 3.. Fid the th term of the followig AP a., 7, 0, 3,. b., 7, 3, -, c. 8,, 0, 9 5 d. 3,, 6,,.... Fid the th term of the followig AP a. 6, 3, 0, 7,. b. 8, 5,,. c. -0, -6, -, d.,, 3,, Fid the umber of terms i the followig AP a. 50, 7,, -3 b. 8,,,., 59 c. -0, -6, -, 6 d. 3, 9, 3, 5,... 39, zefry@sas.edu.my
5 . Give that 6x, 3x ad x + are the first three terms of a AP. Fid a. the value of x, b. the seveth term of the AP. 5. The fifth term of a AP is 3 ad the eleveth term is 5. Fid the fifteeth term. 6. The eighth term ad the eighteeth term of a AP is 37 ad 97 respectively. Fid a. the first term ad the commo differece b. the teth term. zefry@sas.edu.my 5
6 7. Give the AP, -, -5,, 3, Fid the term which is equal to I a AP, the th term, T is give as T = 7 3. Fid a. the first term b. the commo differece c. the 8 th term. Homework : From Text Book- page 0, Exercise., o 3, 5, 8, 0, ad 3. zefry@sas.edu.my 6
7 . 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.. The sum of the first terms of a AP, S is give by i. S a d ii. a l S where l is the last term [ l = T = a + (-)d ]. To determie the th term of ay progressio, T S S Example a. Fid the sum of the first te terms of the AP, 6, 8, 0,. b. Fid the sum of the first 8 terms of the AP, -8, -5, a =, d = S a 0 0 d 8 a = -, d = 3 S a 8 6 d 8 3 zefry@sas.edu.my 7
8 c. Fid the sum of the AP 50, 6,,., Fid first a = 50, d = -, l = T = T = a + ( ) d = 50 + ( )- -36 = - + = 0 = 0 S S 0 a 0 30 d 50 0 d. The sum of the first terms of a AP 7, 0, 3, 6,. is 37. Fid the value of. S a d , () 7 0 (3)( 7) (3) Exercise a. Fid the sum of the first terms of the AP 3, 6, 9. b. Fid the sum of the first 0 terms of the AP -6, -, -6,. c. Give the first three terms of a AP is 7, 0, 3, Fid the sum of the first 6 terms after the fifth term. d. Give the sum of the first th terms of a AP 7, 3, 9,.. is. Fid the value of. zefry@sas.edu.my 8
9 e. Give that the sum of the first terms, S of a AP is give by S = +. Fid i. the first term ii. the commo differece f. I a AP with 5 terms, the fourth term is 8 ad the teth term is. Fid the sum of all the terms i the progressio..5 Solve problems ivolvig AP. I a AP, the fifth term is 6 ad the sum of the first eight terms is 6. Fid the sum of all the terms from the sixth term to the twelfth term. zefry@sas.edu.my 9
10 . Give the first three terms of a AP are 5x-6, 6 x ad 3x. Fid the value of x ad the sum of the first terms. 3. The sum of the first 8 terms of a AP is 8 ad the sum of the ext 8 terms is -6. Fid the a. first term ad the commo differece b. sum of all terms from the 0 th term to the 0 th term.. Give a AP -5, -,,. State the four cosecutive terms i this progressio that sums up to 58. Homework : From Text Book- page, Exercise., o, 6, 7, 8, 0, ad 5 zefry@sas.edu.my 0
11 . GEOMETRIC PROGRESSION. Idetify characteristics of geometric progressio ( GP) GP is a progressio i which each term is multiplied by a costat i order to obtai the ext term. - Example : examples of GP, 8, 3, 8, x x x,,,.. 8. Determie whether a give sequece is a GP Example : Determie whether each of the followig sequeces is a GP a., 3, 9, 7,. b.,,,... 6 It is a GP x 3 x 3 x 3 It is ot a GP Exercise : Determie whether each of the followig sequeces is a GP a., -, 36, -08,.. b.,,, c.,, 9, 6,.. d. 3, 9, 7, 8.. Homework : From Text Book- page, Exercise., o ad. zefry@sas.edu.my
12 .3 Determie by usig formula c. specific terms i GP d. the umber of terms i GP A GP ca be represeted as a, ar, ar, ar 3, ar, First term, a Secod term, T Third term, I geeral, the th term, T is represeted by T 3 Fourth term, T Fifth term, T 5 T = ar where a first term r commo ratio umber of terms Example 3 : a. Fid the eighth term of the GP 3, 6,,. a = 3, r = T = ar - T 8 = (3)() 8- = 38 c. Fid the th term of the followig GP 3, 9, 7 a = 3, r = 3 T = ar - T = (3)(3) - = 3 +(-) = 3 b. Fid the sixth term of the AP,,, a =, r = 3 T = ar - 6 T 6 = 3 = 96 d. Fid the umber of terms i the GP 3, -6,,, -536 a = 3, r= -,. T = -536 T = ar = (3)(-) - -5 = (-) - (-) 9 = (-) - 9 = = 0 zefry@sas.edu.my
13 Exercise 3.. Fid the 7 th term of the followig AP a. 3, 6,,. b., -6, 8, c.,, 8, d.,,, Fid the th term of the followig AP a., 6, 8,. b., 6, 3,. c. 3, 6,,. d.,,, Fid the umber of terms i the followig AP a., 6, 8,. 86 b. 30, 60, 80, 0 c.,,,..., 8 0 d. 3,,,..., 3 3 zefry@sas.edu.my 3
14 . Give that, x ad 9 are the first three terms of a GP. Fid a. the positive value of x, b. the seveth term of the GP. 5. The secod term ad the fourth term of a GP is ad 36 respectively.give that the commo ratio is positive. Fid a. the commo ratio b. the sixth term Give the GP,, 3,9,... Fid the term which is equal to Homework : From Text Book- page, Exercise., o 3,, 5 ad 6. zefry@sas.edu.my
15 . Fid: a) the sum of the first terms 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.. The sum of the first terms of a GP, S is give by a r i. ii. S r, r a r S, r r. To determie the th term of ay progressio, T S S Example a. Fid the sum of the first seve terms of the GP, 8, 6,. a =, r = S a r r 7 S7 508 b. Fid the sum of the first 5 terms of the GP 8, -, 3,.. a = 8, r = a r S r 5 8 S zefry@sas.edu.my 5
16 c. Fid the sum of the GP 5, -0, 0,,30 Fid first a = 5, r = -, l = T = 30 T 30(5)( ) 6 ( ) 6 7 S 7 ar ( ) 6 ( ) 5 5 ( ) 7 d. The sum of the first terms of a GP,,, 8. is 55. Fid the value of. S a r r Exercise a. Fid the sum of the first terms of the GP 3, 6,,. b. Fid the sum of the first 0 terms of the GP 6, -6,, -,. c. Give the first three terms of a GP is 3, 9, 7, Fid the sum of the first 6 terms after the fourth term. d. Give the sum of the first terms of a GP, 8, 6,.. is 508. Fid the value of. zefry@sas.edu.my 6
17 e. Give that the sum of the first terms, S of a GP is give by S = ( ). Fid i. the first term ii. the commo ratio. f. Give that the third term of a GP is 0 ad the eighth term is 60. Fid the sum of the first te terms. Homework : From Text Book- page 3, Exercise., o, 6 ad 8. zefry@sas.edu.my 7
18 .5 Fid: a) the sum to ifiity of geometric progressios. b) the first term or commo ratio, give the sum to ifiity of geometric progressios. The sum to ifiity of a GP whe - < r < is give by S a r Example 5. Fid the sum to ifiity for the followig GP a. 6, 8,,. a = 6, r = b. 7, -9, 3,. a = 7, r = 3 S a r 6 3 S a r Express the recurrig decimal as a fractio i its lowest term a = a = 0.5, r = b = a = 0.36, r = zefry@sas.edu.my 8
19 Exercise 5. Fid the sum to ifiity for the GP,,, Fid the sum to ifiity for the GP 7, 9, 3,. 3. Fid the sum to ifiity for the GP 7, 9,,.... Express the recurrig decimal 0.55 as a fractio i its lowest term 5. Express the recurrig decimal as a fractio i its lowest term 6. Express the recurrig decimal as a fractio i its lowest term zefry@sas.edu.my 9
20 7. A GP is give 6, 3, 3, Fid the sum to ifiity of this progressio 8. I a GP, the first term is 5 ad the fourth term is 8. Fid a. the commo ratio b. the sum to ifiity of this progressio. j. Give that the sum to ifiity of a GP is 0 ad the first term is 0. Fid the commo ratio Homework : From Text Book- page, Exercise., o 0,,, 3, ad 7 zefry@sas.edu.my 0
21 .6 Solve problems ivolvig GP. The secod term of a GP is 6 ad the sum of the first two terms is 9. Fid the i. commo ratio ad the first term ii. the sum of all the terms from the fourth term to the eighth term.. Fid the least umber of terms i the GP,, 36,. that is greater tha zefry@sas.edu.my
22 8 3. I a GP, the fifth term is ad the sum of the third term ad fourth term is 9. Give that r < 0, fid ii. the commo ratio iii. the first term iv. the sum of the first 6 terms. Homework : From Text Book- page 6, Chapter Review, o 5, 6, 7 ad 8 zefry@sas.edu.my
23 PAST YEAR QUESTIONS Paper. It is give that the first four terms of a GP are 3, -6, ad x. Fid the value of x.. The first three terms of a AP are 6, 3 ad 0. The th term of this progressio is egative. Fid the least value of. 3. a) Determie whether the followig sequece is a AP or a GP 6x, 8x, x, b) Give the reaso for the aswers i 9(a).. Three cosecutive terms of a AP are 5 x, 8, x. Fid the commo differece of the progressio. 5. The first three terms of a GP are 7, 8,. Fid the sum to ifiity of the GP. 6. The ith term of a AP is + 5p ad the sum to ifiity of the first four terms of the progressio is 7p 0, where p is a costat. Give that the commo differece of the progressio is 5, fid the value of p. 7. The third term of a GP is 6. The sum of the third term ad the fourth term is 8. Fid a. the first term ad the commo ratio of the progressio b. the sum to ifiity of the progressio. 8. The first three terms of a sequece are, x, 8. Fid the positive value of x so that the sequece is a. a arithmetic progressio b. a geometric progressio 9. 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 0 terms after the third term. 0. The sum of the first terms of a GP 8,, 7, is 87. Fid a. the commo ratio of the progressio b. the value of.. Give a GP y,,, p..., express p i terms of y. 5. Give a AP -7, -3,,, state three cosecutive terms i this progressio which sum up to 75. zefry@sas.edu.my 3
24 3. Express the recurrig decimal as a fractio i its simplest form.. The first three terms of a arithmetic progressio are k 3, k + 3, k +. Fid a. the value of k b. the sum of the first ie terms of this progressio. 5. I a GP, the first term is 6 ad the fourth term is 7. Calculate a. the commo ratio b. the sum to ifiity of the GP Paper. Muthu started workig for a compay o Jauary 00 with a iitial aual salary of RM Every Jauary, the compay icreased his salary by 5% of the previous year s salary. Calculate a. his aual salary, to the earest RM, for the year 007 b. the miimum value of such that his aual salary i the th year will exceed RM c. the total salary, to the earest RM, paid to him by the compay, for the years 00 to Diagram shows the side elevatio of part of stairs built of cemet blocks. 95 cm 955 cm 985 cm 5 cm Diagram The thickess of each block is 5 cm. The legth of the first block is 985 cm. The legth of each subsequet block is 30 cm less tha the precedig block as show i diagram. a. If the height of the stairs to be built is 3 m, calculate i. the legth of the top most block zefry@sas.edu.my
25 ii. the total legth of the blocks b. Calculate the maximum height of the stairs. 3. Two compaies, Delta ad Omega, start to sell cars at the same time. a. Delta sells k cars i the first moth ad its sale icreases costatly by m cars every subsequet moth. It sells 0 cars i the 8 th moth ad the total sales for the first 0 moths are 900 cars. Fid the value of k ad of m. b. Omega sells 80 cars i the first moth ad its sales icrease costatly by cars every subsequet moth. If both compaies sell the same umber of cars i the th moth, fid the value of.. Diagram shows part of a arragemet of bricks of equal size. 6 cm Diagram The umber of bricks i the lowest row is 00. For each of the other rows, the umber of bricks is less tha i the row below. The height of each brick is 6 cm. Ali builds a wall by arragig bricks i this way. The umber of bricks i the highest row is. Calculate a. the height, i cm, of the wall b. the total price of the bricks used if the price of oe brick is 0 se. 5. Diagram 3 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 Diagram 3 x cm zefry@sas.edu.my 5
26 a. Show that the areas of the triagles form a geometric progressio ad state the commo ratio. b. Give that x = 80 cm ad y = 0 cm, i. Determie which triagle has a area of 6 cm. ii. fid the sum to ifiity of the areas, i cm, of the triagles. ANSWERS Paper a. GP b. the ratio of two cosecutive terms of the sequece is a costat a a = 6 r b a. AP b. GP 9. a. b a. 3 b. 7. p 8 3., 5, 9 3. y 33. a. 7 b a. 3 b. 56 Paper. a. RM 973 b. = 6 c. RM 3. a. i. 5 cm ii. 000 cm b. 95 cm 3. a. m = 0, k = 00 b.. a. 9 cm b. RM a. Triagles form a GP with commo ratio = b. i. 5 th triagle ii zefry@sas.edu.my 6
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