ARITHMETIC PROGRESSION

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1 CHAPTER 5 ARITHMETIC PROGRESSION Poits to Remember :. A sequece is a arragemet of umbers or objects i a defiite order.. A sequece a, a, a 3,..., a,... is called a Arithmetic Progressio (A.P) if there exists a costat d such that a a = d, a 3 a = d, a 4 a 3 = d,..., a a = d ad so o. The costat d is called the commo differece. 3. If a is the first term ad d the commo differece of a A.P., the the A.P. is a, a + d, a + d, a + 3d The th term of a A.P. with first term a ad commo differece d is give by a a ( ) d. 5. The sum to terms of a A.P. with first term a ad commo differece d is give by S [ a ( ) d] Also, S [ a l], where l = last term. ( ) 6. Sum of first atural umbers = Example. ILLUSTRATIVE EXAMPLES A sequece a is defied by a = 5. Prove that it is a AP. Solutio. Give a 5 a 5 ( ) 5 7 Cosider, a a (5 ) ( 7) = =. We observe that a a is idepedet of ad hece a costat. The give sequece is a A.P. with commo differece. Example. Which term of the A.P. 5, 5, 5,... will be 30 more tha its 3 st term? [CBSE 006(C)] Solutio. We have, a 5, d 0 a3 a (3) d a 30d Let th term of the give A.P. be 30 more tha its 3 st term. The, a = 30 + a 3 a + ( ) d = ( ) = ( ) = 430 = 43 = 44. Hece, 44 th term of the give A.P. is 30 more tha its 3 st term. Example 3. Which term of the A.P. 9, 8,7,... is the first egative term? 5 5 Solutio. 4 Here, a 9, d MATHEMATICS X ARITHMETIC PROGRESSIONS 7

2 Let the th term of the give A.P. be the first egative term. i.e. a + ( ) d < ( ) < 0 99 < The smallest atural umber satisfyig 4 is 5. 4 The first egative term of give A.P. is 5th term. Example 4. How may umbers of two digits are exactly divisible by 8? Solutio. We observe that 6 is the first two digit umber divisible by 8 ad 96 is the last two digit umber divisible by 8. Thus, we have to determie the umber of terms i the sequece 6, 4, 3,..., 96. Clearly, it is a A.P. with first term = 6 ad commo differece = 8, i.e. a = 6, d = 8. Let there be terms i this A.P. The, a = 96 a + ( ) d = ( ) (8) = 96 8 ( ) = 80 = 0 = Hece, there are umbers of two digits which are divisible by 8. Example 5. Fid the sum of the first 5 terms of a A.P. whose th term is give by a = 7 3. [CBSE 004] Solutio. We have, a 7 3 Puttig =, we get a 7 3() 4 Puttig = 5, we get a5 7 3(5) 68 so, for the give A.P., we have, first the = a = 4 ad 5th term = 68. Required sum = 5 S [ ] 5 a a ( ) 5 S a l 5 (4 68) 5 ( 3) 800 As. Example 6. How may terms of the series 54, 5, 48,... be take so that their sum is 53? [CBSE 005] Solutio. Clearly, the give sequece is a A.P. with first term a ( = 54) ad commo differece d (= 3). Let the sum of terms be 53. The, S 53 [ a ( ) d] 53 [08 ( )( 3)] = 0 ( 8) ( 9) = 0 = 8 or = 9 Here, the commo differece is egative. So, 9th term is give by a9 54 (9 ) ( 3) 0. Thus, the sum of 8 terms as well as that of 9 terms is ARITHMETIC PROGRESSIONS MATHEMATICS X

3 Example 7. Solutio. Example 8. Solutio I a A.P., the sum of first terms is Fid its 5th term. [CBSE 006(C)] Let S deote the sum of terms of a A.P. whose th term is a. We have, S S ( ) ( ) [ replacig by ( )] a S S ( ) ( ) 3 5 [ ( ) ] [ ( )] 3 5 ( ) 3 5 a5 ( 5 ) 76 As. If the sum of the first 0 terms of a A.P. is 0 ad the sum of first 0 terms is 40, fid the sum of first 30 terms of this A.P. Let the first term be a ad the commo differece be d. Now, Give S0 0 0 [ a (0 ) d] 0 a + 9d =...() Also, S 0 = 40 0 [ a (0 ) d] 40 a + 9d = 4...() Subtractig () from (), we get 0d = 0 d = Usig d = i (), we get a + 9 () = a = Now, S [ a ( ) d] 30 S 30 [() (30 )()] = 5 6 = 930 As. Example 9. A cotract o costructio job specifies a pealty for delay of completio beyod a certai date as follows : Rs. 00 for the first day, Rs. 50 for the secod day, Rs. 300 for the third day, etc. the pealty for each succeedig day beig Rs. 50 more tha for the precedig day. How much moey the cotractor has to pay as pealty, if he has delayed the work by 30 days? [NCERT] Solutio. Clearly, the amout of pealty for differet days forms a A.P. with first term a (= 00) ad commo differece d (=50). Accordig to questio, we have to fid the sum of 30 terms of this A.P. MATHEMATICS X ARITHMETIC PROGRESSIONS 73

4 Now, S [ a ( ) d] here, a 00, d 50, Required sum S30 [(00) (30 )(50)] =5 ( ) = = 7750 Thus, a delay of 30 days will cost the cotractor of Rs. 7,750. Example 0. A spiral is made up of successive semicircles, with cetres alteratively at A ad B, startig with cetre at A, of radii 0.5 cm,.0 cm,.0 cm, as show i figure. What is the total legth of such a spiral made up of thirtee cosecutive semicircles? l 3 l A B Solutio. r = 0.5 cm, r =.0 cm, r 3 =.5 cm,... here, a = 0.5, d = = 0.5 legth of spiral made up of 3 cosecutive semicircles r r... r 3 ( r r... r3 ) = ( upto 3 th term) 3 [(0.5) (3 )(0.5)] l Take 7 [NCERT] cm As. 7 Example. 00 logs are stacked i the followig maer : 0 logs i the bottom row, 9 i the ext row, 8 i the row ext to it ad so o (see diagram). I how may rows the 00 logs are placed ad how may logs are i the top row? [NCERT] Solutio. Here, a = 0, d =, S = 00 Now, a = 0 a + ( ) d = 0 a + ( ) () = 0 a + =...() 74 ARITHMETIC PROGRESSIONS MATHEMATICS X

5 Also, S [ a a ] 00 [ a 0] a + 0 = () from () ad (), we get ( ) + 0 = 400 ( a ) = 0 ( 6) ( 5) = 0 = 6 or = 5 ` (i) Takig = 5, we get a + ( ) d = 0 a + (5 ) () = 0 a = 4, which is ot possible as umber of logs caot be egative. = 5 is rejected. (ii) Takig = 6, we get a + ( ) d = 0 a + (6 ) () = 0 a = 5 Number of logs i the first row = 5, ad the umber of rows = 6 As. Example. I a potato race, a bucket is placed at the startig poit, which is 5 m from the first potato, ad the other potatoes are placed 3 m apart i a straight lie. There are te potatoes i the lie (see diagram). Fid the total distace travelled for placig the te potatoes from a lie ito the bucket. Solutio. Clearly, the competitor is to cover double distace i order to get the potato from the place ad to be placed i the bucket. Distace betwee bucket ad I st potato = 5 m Distace betwee bucket ad d potato = (5 + 3) m = 8 m Distace betwee bucket ad 3 rd potato = (8 + 3) m = m here, a = 5, d = 8 5 = 3 Total distace travelled for placig 0 potatoes i the bucket = ( upto 0 terms) 0 (5 (0 )3) = [5 (0 + 7)] = 370 m As. MATHEMATICS X ARITHMETIC PROGRESSIONS 75

6 Example 3. A ladder has rugs 5 cm apart (see figure). The rugs decrease uiformly i legth from 45 cm Solutio. at the bottom ad 5 cm at the top. If the top ad the bottom rugs are m apart, what is the legth of the wood required for the rugs? [NCERT] Distace betwee rug at the top ad the bottom m=50 cm. Distace betwee two cosecutive rugs = 5 cm. 50 Number of rugs required 0. 5 Legth of rugs is icreasig uiformly ad formig a A.P. a = 5, a = 45 Legth of wood required to make rugs 5 cm cm m S [ ] 5 cm a l 45 cm [5 45] 385 cm 3.85 m As. Example 4. The houses of a row are umbered cosecutively from to 49. Show that there is a value of x such that the sum of the umbers of the houses precedig the house umber x is equal to the sum of the umbers of the houses followig it. Fid this value of x. [NCERT] Solutio. Let the umber of houses before house o. x = x. 49 houses 3... x x x+ x ( x ) houses (49 x) houses Sice, house umbers are cosecutive, so sum of house umbers precedig house x x( x ) 3... ( x ) Now, the umber of houses followig x = 49 x So, sum of house umbers followig x = (x + ) + (x + ) x [( x ) (49 x )()] 49 x (50 x) x( x ) (49 x) Accordig to the give questio, (50 x ) x x = 49 (50 + x) x (50 + x) x = x = 49 5 = 7 5 = (7 5) = (35) x = 35 As. ( ) S 76 ARITHMETIC PROGRESSIONS MATHEMATICS X

7 Example 5. A small terrace at a football groud comprises of 5 steps each of which is 50 m log ad built Solutio. of solid cocrete. Each step has a rise of 4 m ad a tread of m (see figure). Calculate the total volume of cocrete required to build the terrace. [NCERT] Volume of cocrete eeded to make first step lbh 50 m = m 4 4 Volume of cocrete eeded to make secod step m = m 4 4 Volume of cocrete eeded to make third step MATHEMATICS X ARITHMETIC PROGRESSIONS 77 m m = m m here, a, d Total volume of cocrete eeded to make such 5 steps (5 ) m As. 4 Problem based o th term of a A.P. PRACTICE EXERCISE Determie which of the followig are A.Ps. If they form a AP, fid the commo differece d ad write three more terms (Qs. -6) :. 0, 4, 8,,....,, 4, 8, 6, , 6, 6, 6, 6, , 5 3, 5 3, 5 3 3, , 0 + 5, 0 + 5, ,... 6., 8, 8, 3, Fid the sequece whose of th term is give by : (a) 7 (b) (c) (d) Also, determie which of these sequeces are A.P s. 8. Fid the A.P. whose th term is give by : (a) 9 5 (b) + 6 (c) Fid the 8 th term of a A.P. whose 5 th term is 47 ad the commo differece is The 0 th term of a A.P. is 5 ad 6 th term is 8. Fid the 3 d term ad the geeral term.. The 7 th term of a A.P. is 4 ad its 3 th term is 6. Fid the A.P. [CBSE 004] 50m

8 . Which term of the A.P. 3, 0, 7,... will be 84 more tha its 3 th term? [CBSE 004] 3. Fid the A.P. whose third term is 6 ad the seveth term exceeds its fifth term by. 4. For what value of is the th term of the followig A.P. s the same?, 7, 3, 9,... ad 69, 68, [CBSE 006C] 5. If the 0 th term of a A.P. is 5 ad 7 th term is 0 more tha the 3 th term, fid the A.P. 6. If seve times the seveth term of a A.P. is equal to ie times the ith term, show that 6th term is zero. 7. (a) Which term of the A.P., 6, 0, 4,... is 78? (b) Which term of the A.P. 5, 9, 3, 7,... is 5? (a) Which term of the A.P. 4,,, 5,... is 9? (a) Which term of the A.P., 8, is? 8. How may terms are there i each of the followig fiite A.P. s? (a) 3, 4, 5, 6,..., 07 (b) 7, 3, 9,..., (c) 8, 3, 8,..., 08 (d) 0, 9,9,...,6 9. (a) Is 6 a term of A.P. 3, 8, 3, 8,...? (b) Is 7 a term of A.P., 4, 7, 0,...? (c) Is 5 a term of A.P. 0, 7, 4...? (d) Is 90 a term of A.P. 3, 8, 3,...? 0. Fid the 8th term from the ed of the A.P. 7, 0, 3,..., 84. [CBSE 005]. Fid the 0th term from the ed of the A.P. 3, 8, 3,..., 53.. Fid the value of k so that k 3, 4k ad 3k 7 are three cosecutive terms of a A.P. 3. Fid the value of x so that 3x +, 7x ad 6x + 6 are three cosecutive terms of a A.P. 4. The 4 th term of a A.P. is three times the first ad the 7 th term exceeds twice the third term by. Fid the first term ad the commo differece. 5. The 9 th term of a A.P. is equal to 7 times the d term ad th term exceeds 5 times the 3 rd term by. Fid the first term ad the commo differece. 6. Fid the middle term of a A.P. with 7 terms whose 5 th term is 3 ad the commo differece is. 7. For what value of, the th terms of the two A.P. s are same?, 3, 8, 3,... ad 6, 7, 8, 9, For what value of, the th terms of the two A.P. s are same? 3, 0, 7,... ad 63, 65, 67, Fid the umber of itegers betwee 00 ad 500 which are divisible by How may umbers of three digits are exactly divisible by? 3. The agles of a triagle are i A.P. If the greatest agle equals the sum of the other two, fid the agles. 3. Three umbers are i A.P. If the sum of these umbers is 7 ad the product is 648, fid the umbers. 33. If m times the m th term of a A.P. is equal to times its th term, show that the (m + ) th term of the A.P. is zero. 34. A sum of Rs. 000 is ivested at 8% simple iterest per aum. Calculate the iterest at the ed of,, 3,...years. Is the sequece of iterest a A.P.? Fid the iterest at the ed of 30 years. 35. Two A.P. s have the same commo differece. The differece betwee their 00th terms is 333. What is the differece betwee their millioth terms? 78 ARITHMETIC PROGRESSIONS MATHEMATICS X

9 Problems based o sum to terms of a A.P. (S ) : 36. Fid the sum of first atural umbers. Fid the sum of the followig series (Qs ) : to 50 terms to 0 terms to 0 terms upto terms. [CBSE 007 (C)] 4. Fid the sum of first 30 eve atural umbers. 4. Fid the sum of first 5 odd atural umbers. 43. Show that the sum of the first odd atural umbers equals. 44. Fid the sum of terms of a A.P. whose th term is give by a = I a A.P., a = 6, a = ad d = 6. Fid S. 46. The th term of a A.P. is give by t = 4 5. Fid the sum of the first 5 terms of the A.P. [CBSE 004] Fid the sum of all the atural umbers (Qs ) : 47. betwee 50 ad 000 which are exactly divisible by betwee 50 ad 500 which are divisible by betwee 00 ad 000 which are multiples of betwee 50 ad 500 which are divisible by 3 ad How may terms of the sequece 35, 8,,... should be take so that their sum is zero? 5. How may terms of the A.P. 63, 60, 57,... must be take so that their sum is 693? [CBSE 005] 53. How may terms of a A.P., 4, 7,... are eeded to give the sum 335? 54. How may terms of a A.P. 6,, 5... are eeded to give the sum 5? Explai the double aswer. Fid the followig sum (Qs ) : ( 5) + ( 8) + ( ) ( 30) I a A.P., the sum of the first terms is. Fid its 5th term. [CBSE 006 C] 6. If the sum of the first 6 terms of a A.P. is 36 ad the sum of the first terms is 44, fid the first term ad the commo differece ad hece fid the sum of terms of this A.P. 63. Fid the commo differece of a A.P. whose first term is ad the sum of the first four terms is oe-third the sum of the ext four terms. 64. The sum of the first 9 terms of a A.P. is 7 ad that of the first 4 terms is 996, fid the first term ad the commo differece. MATHEMATICS X ARITHMETIC PROGRESSIONS 79

10 65. If the sum of the first 0 terms of a A.P. is 400 ad the sum of the first 40 terms is 600, fid the sum of its first 0 terms. 66. Fid the sum of first 0 terms of a A.P., i which 3 rd term is 7 ad 7 th term is two more tha thrice of its 3 rd term. 67. The sums of -terms of two A.P. s are i the ratio 7 + : Fid the ratio of their th terms. 68. (a) For a A.P., fid S if give a = 3 5. (b) For a A.P., fid a if give S = A perso borrows Rs ad promises to pay back (without ay iterest) i 30 istalmets each of value Rs. 0 more tha last (precedig oe). Fid the first ad the last istalmets. 70. A ma saved Rs. 3 durig the first year, Rs. 36 i the secod year ad i this way be icreases his savigs by Rs. 4 every year. Fid i what time his savigs will be Rs Fid the three umbers i A.P. such that their sum is 4 ad the sum of their squares is The sum of the 4th ad 8th terms of a A.P. is 4 ad the sum of the 6th ad 0th terms is 44. Fid the sum of first 0 terms of the A.P. 73. The sum of the third ad the seveth terms of a A.P. is 6 ad their product is 8. Fid the sum of first te terms of the A.P. 74. If the m th term of a A.P. be ad th term be, m the show that its (m)th term is. 75. The 0 th term of a A.P. is 9 ad ad sum of the first 0 terms is 60. Fid the sum of the first 30 terms. 76. The sum of the first 5 terms of a A.P. is 05 ad the sum of ext 5 terms is 780. Fid the first three terms of the A.P. 77. If mth term of A.P. is ad th term is, m show that the sum of m terms is ( ). m 78. Two cars start together i the same directio from the same place. The first goes with a uiform speed of 0 km/hr. The secod goes with a speed of 8 km/hr i the first hour ad icreases the speed by km/hr i each succeedig hour. After how may hours will the secod car overtake the first if both cars go ostop? 79. The ages of the studets i a class are i A.P. whose commo differece is 4 moths. If the yougest studet is 8 years old ad the sum of the ages of all the studets is 68 years, fid the umber of studets i the class. 80. I a A.P. (i) give a = 5, d = 3, a = 50, fid ad S. (ii) give a = 7, a 3 = 35, fid d ad S 3. (iii) give a = 37, d = 3, fid a ad S. (iv) give a 3 = 5, S 0 = 5, fid d ad a 0. (v) give d = 5, S 9 = 75, fid a ad a 9. (vi) give a =, d = 8, S = 90, fid ad a. (vii) give a = 8, a = 6, S = 0, fid ad d. (viii) give a = 4, d =, S = 4, fid ad a. (ix) give a = 3, = 8, S = 9, fid d. (x) give l = 8, S = 44, ad there are total 9 terms. Fid a. 80 ARITHMETIC PROGRESSIONS MATHEMATICS X

11 HINTS TO SELECTED QUESTIONS 5. Let first term be a ad commo differece be d. The, a9 7a a 8d 7( a d) ad, a 5a3 ( a d) 5( a d). 3. Let the agles be a d, a, a + d. The, (a d) + a + (a + d) = 80 3a = 80 a = 60 agles are 60 d, 60 ad 60 + d. Also, 60 + d = (60 d) + 60 d = 30 agles are 30, 60 ad a 5 6 for =, a = a = 5 6() = Now, S [ a a ] [ 5 6 ] ( 6 4) ( 3 ) = 3 6. here, S 3 3. Now, a S S a5 S5 S4 3(5) 3(5) 3(4) 3(4) [3 49 3] here, a =. Let commo differece be d. Let A.P. be a, a, a 3, a 4, a 5, a 6,... The, a + a + a 3 + a 4 ( ) 3 a a a a [3(5 4 ) 3(5 4)] 3 (a + a + a 3 + a 4 ) = a 5 + a 6 + a 7 + a 8 3 (a + a + a 3 + a 4 ) = (a + a a 7 + a 8 ) (a + a a 4 ) 3 S 4 = S 8 S 4 4S 4 = S [() (4 ) d] [() (8 ) d] d = 67. Let a, a be first terms ad d, d the commo differeces of the two give A.P. s. The, [ a ( ) d] a d S a ( ) d S ( ) [ a ( ) d ] a d a d MATHEMATICS X ARITHMETIC PROGRESSIONS 8

12 S 7 Give, S 4 7 a d a d a Clearly, a 0d a a 0d Comparig () ad (), we get 0 a a 7() 4 : 3. 4() 7...()...() 77. Let a be the first term ad d be the commo differece of the give A.P. The, a m a ( m ) d...() a a ( ) d m m...() Solvig () ad (), we get a d m m Now, Sm [ a ( m ) d] Put a d i S m m ad solve. 78. Let the secod car overtakes the first car after t hours. The, the two cars travel the same distace i t hours. Now, Distace travelled by the secod car = sum of t terms of a A.P. with a = 8, d. t( t 3) 4 t( t 3) accordig to questio, 40 t t 9t 0 t = 9 [ t 0] here, a = 8, d, S 68 4 The, S [ a ( ) d ] 68 (8) ( ) 4 Simplify ad get quadratic equatio i. 8 ARITHMETIC PROGRESSIONS MATHEMATICS X

13 MULTIPLE CHOICE QUESTIONS Mark the correct alterative i each of the followig :. Which of the followig sequece whose th term is give by a, is ot a A.P. : (a) a = 3 (b) a = 3 7 (c) a = + (d) a = The 3 d term of a AP, whose 0 th term is 5 ad 6 th term is 8, is: (a) 6 (b) 5 (c) 7 (d) oe of these 3. I m th term of a AP is, ad th term is, m the its (m)th term is : m (a) 0 (b) (c) m 4. The total umber of terms i a A.P. 8,5,3,..., 47 are : (d) m m (a) 0 (b) 5 (c) 7 (d) The total umber of multiples of 4, betwee 0 ad 50 are : (a) 45 (b) 50 (c) 60 (d) oe of these 6. If th term of a A.P. is ( + ), the the sum of first terms of the A.P. is : (a) ( ) (b) ( ) (c) ( 3) 7. The sum of all three digit atural umbers, which are divisible by 7, is : (d) oe of these (a) (b) (c) (d) oe of these 8. The total umbers of terms of A.P. 9, 7, 5,... that must be take so that their sum is 636, is : (a) 0 (b) (c) (d) 3 9. If the sum of terms of a A.P. is + 5 the its th term is : (a) 4 3 (b) 3 4 (c) (d) If the first term of a A.P. is ad commo differece is 4, the the sum of its 40 terms is : (a) 300 (b) 600 (c) 000 (d) 3000 VERY SHORT ANSWER TYPE QUESTIONS ( MARK QUESTIONS). Fid a, such that 5, a, 35 are i A.P.. Fid the 0th term of a A.P. 3,,, Write the 5th term of a A.P. whose first term is 7 ad the commo differece is Write the missig terms of the A.P. 9,, 9, 4, 5. For what value of k, the umbers k +, 4k + 4 ad 9k + 4 are three cosecutive terms of a A.P. MATHEMATICS X ARITHMETIC PROGRESSIONS 83

14 6. Write the first three terms of a A.P. whose th term is Write the A.P. whose th term is Which term of the A.P., 6, 0,... is 0? 9. Is 67 a term of the A.P. 7, 0, 3,...? 0. Fid the umber of terms of A.P. 7, 3, 9,..., 30.. The 7 th term of a A.P. exceeds its 0 th term by 7. Fid the commo differece.. How may -digit umbers are divisible by 3? 3. Which term of the A.P. 9, 8, 7,... is the first egative term? Fid the sum of first 50 atural umbers. 5. Fid the followig sum : terms. 6. If the sum of first terms of a A.P. is + 5, write the sum of its first 5 terms. 7. Fid the sum of first 5 terms of a AP, whose th term is Give that the first term of a A.P. is ad its commo differece is 4, fid the sum of its first 40 terms. 9. Fid the sum of the odd umbers betwee 0 ad Fid the sum to terms of the A.P. whose rth term is 5r +. M.M : 30 Geeral Istructios : PRACTICE TEST Q. -4 carry marks, Q. 5-8 carry 3 marks ad Q. 9-0 carry 5 marks each.. Fid the term of A.P. 9,, 5, 8,... which is 39 more tha its 36 th term.. Fid the umber of itegers betwee 50 ad 500 which are divisible by Fid the sum of first 5 multiples of Fid the sum of first 5 terms of a A.P. whose th term is give by a = If the sum of 7 terms of a A.P. is 49 ad that of 7 terms is 89, fid the sum of terms. Time : hour 6. If the sum of m terms of a A.P. is the same as the sum of its terms, show that the sum of its (m + ) terms is zero. 7. Prashi saved Rs. 5 i the first week of the year ad the icreased her weekly savigs by Rs..75 each week. I what week will her weekly savigs be Rs. 0.75? 8. The sum of the third ad the seveth terms of a A.P. is 6 ad their product is 8. Fid the sum of first sixtee terms of the A.P. 9. (a) Fid the th term from the ed of the A.P. 3, 8, 3,..., 53. (b) How may three digit umbers are divisible by 7? 0. Salma Kha buys a shop for Rs., 0,000. He pays half of the amout i cash ad agrees to pay the balace i aual istalmets of Rs each. If the rate of iterest is % ad he pays with the istalmet the iterest due o the upaid amout, fid the total cost of the shop. 84 ARITHMETIC PROGRESSIONS MATHEMATICS X

15 ANSWERS OF PRACTICE EXERCISE. Yes, d = 4; 6, 0, 4. No 3. Yes, d = 0; 6, 6, 6 4. Yes, d 3;5 4 3, 5 5 3, No 6. Yes, d ; 50, 7, (a) ad (c) form a A.P. 8. (a) 4,, 6,,... (b) 5, 4, 3,,... (c) 9,, 3, 5, a 3 = 6, a = , 6, 4, th 3. 4, 0, 6, No value of 5. 7,, 7, (a) 0 th term (b) 3 st term (c) 4 d term (d) th term 8. (a) 05 (b) 35 (c) 4 (d) 9 9. (a) No (b) Yes (c) No (d) No k = 3 3. x = 4. a = 3, d = 5. a =, d = = 8 8. = , 60, , 9, 34. Rs. 80, Rs. 60, Rs. 40, ad so o. Yes; Rs ( ) or or a =, d = ; S = a = 7, d = : (a) ( 5) (b) Rs. 5, Rs years. 7. 7, 8 ad or ,, hours (i) = 6, S = 440 (ii) d 7, S (iii) a 4, S (iv) d, a0 8 (v) a, a9 3 3 (vi) = 5, a = (vii) 6, d 5 (viii) a = 8, = 7 (ix) d = 6 (x) a = 4 MATHEMATICS X ARITHMETIC PROGRESSIONS 85

16 ANSWERS OF MULTIPLE CHOICE QUESTIONS. (c). (a) 3. (b) 4. (c) 5. (c) 6. (b) 7. (a) 8. (c) 9. (c) 0. (a). a = ANSWERS OF VERY SHORT ANSWER TYPE QUESTIONS , 9 5. k = 6.,, ,, 5, rd 9. yes th (5 7) ANSWERS OF PRACTICE TEST. 49 th th week 8. 0 or (a) 98 (b) 8 0. Rs.,66, ARITHMETIC PROGRESSIONS MATHEMATICS X

ARITHMETIC PROGRESSIONS

CHAPTER 5 ARITHMETIC PROGRESSIONS (A) Mai Cocepts ad Results A arithmetic progressio (AP) is a list of umbers i which each term is obtaied by addig a fixed umber d to the precedig term, except the first