Each term is formed by adding a constant to the previous term. Geometric progression

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1 Chpter 4 Mthemticl Progressions PROGRESSION AND SEQUENCE Sequence A sequence is succession of numbers ech of which is formed ccording to definite lw tht is the sme throughout the sequence. Arithmetic Progression Ech term is formed by dding constnt to the previous term. Geometric progression Ech term is formed by multiplying the previous term by constnt. ARITHMETIC PROGRESSION An rithmetic progression is one where ech term in the sequence is linked to Definition the immeditely preceding term by dding or subtrcting constnt number. The number dded or subtrcted to construct the progression is known s the Common common difference. In other words there is common difference between difference d ech pir of successive numbers in the sequence. Sequence Common difference 0,,, 3, 4, 5, 6 etc. + 0, 3, 6, 9,, 5 etc. +3 5, 0, 5, 0, 5, 0, -5 etc. -5 Note tht for three consecutive numbers w, x, y nd z in n rithmetic AP property progression: nd term st term = 3 rd term nd term First term Number of terms n Vlue of nth term Sum of number of terms The first term in the series is expressed by constnt The totl number of terms in the series under review is expressed by constnt n nth term = + (n )d n [ + (n ) d] Or n ( + l) where l is lst term in the series Form qudrtic eqution on the bsis of d, nd s dn + [( ) d]n + (s ) = 0 Solving for n Solve qudrtic eqution for n n + bn + c = 0 n = b ± b 4c Pge of 7 (kshifdeel.com)

2 QUESTION BANK nth term (AP) 0 7 th term of n A.P. 8,5,, -, -4. Is () 9 (b) 0 (c) -9 (d) -0 ANSWER 0 nth term = + (n )d = 8, d = 3, n = 7 7th term = 8 + (7 ) 3 7th term = 0 0 Which term of the AP 3, 4, 5 7.is () 5 (b) 7 (c) 6 (d) 8 ANSWER 0 nth term = + (n )d Ignore 7 7 = 3 + (n ) = 3, d =, nth term = 7 n = 5 03 An uditorium hs 0 sets in the front row, 5 sets in the second row, nd 30 sets in the third row nd so on for 3 rows. Number of sets in the 3th row re: () 70 (b) /3 (c) 8 (d) 3/ ANSWER 03 nth term = + (n )d = 0, d = +5, n = 3 3th term = 0 + (3 ) + 5 3th term = 80 Pge of 7 (kshifdeel.com)

3 QUESTION BANK AP PROPERTIES 04 ( b),, ( + b) re in progression () Geometric (b) Arithmetic (c) Hrmonic (d) None of theses ANSWER 04 ( b) = ( + b) + b = + b Prove tht it is n AP nd term first term = 3 rd term nd term 05 The vlue of x such tht 8x + 4, 6x -, x + 7 will form n AP is: () 5 (b) (c) 5/ (d) None of these ANSWER 05 (6x ) (8x + 4) = (x + 7) (6x ) 6x 8x 4 = x + 7 6x + 6x 8x x + 6x = x = 5 nd term st term = 3 rd term nd term Brckets opened Solved for x x = 5 06 If the terms x,(x + 0 ) nd (3x + ) be in A.P., the vlue of x is; () 7 (b) 0 (c) 6 (d) None of these ANSWER 06 (x + 0) x = (3x + ) (x + 0) x + 0 x = 3x + x 0 x x 3x + x = x = 8 x = 6 nd term st term = 3 rd term nd term Brckets opened Solved for x Pge 3 of 7 (kshifdeel.com)

4 QUESTION BANK AP SUM 07 The sum of the series n is () (n + )/ (b) n( n - )/ (c) n( n + )/ (d) None of these ANSWER 07 =, d = Dt n [ + (n ) d] Formul n [() + (n ) ()] n [ + n ] Simplified n [n + ] Simplified Avilble vlues = (),500 (b),550 (c),575 (d) None of these ANSWER 08 n [ + (n ) d] =, d =, n = [() + (50 ) ], The sum of ll odd numbers between 00 nd 300 is: (),600 (b),490 (c),500 (d) 4,750 ANSWER n [ + (n ) d] = 0, d =, n = 50 [(0) + (50 ) ],500 0 The first nd the lst term of n AP re -4 nd 46. The sum of the terms is 77. The number of the terms is: () 0 (b) 00 (c) 99 (d) None of these ANSWER 0 n ( ) = 4, l = 46, = n ( + l) n = 0 Pge 4 of 7 (kshifdeel.com)

5 QUESTION BANK AP (Solving for n ) The sum of certin number of terms of n AP series -8, -6, -4. Is 5 the number of terms is: () (b) 3 (c) (d) None of these ANSWER dn + [( ) d]n + (s ) = 0 = 8, d = +, 5 n + [( 8 ) ]n + (5 ) = 0 n 8n 04 = 0 n = ( 8) ± ( 8) 4 ( 04) n = 3 or n = 4 Number of terms cnnot be in negtive A person hs to py Rs.975 by monthly instllments ech less then the former by Rs.5. the first instllment is Rs. 00. The time by which the entire mount will be pid is: () 0 months (b) 5 months (c) 4 months (d) None of these ANSWER dn + [( ) d]n + (s ) = 0 = 00, d = 5, 975 5n + [(00 ) ( 5)]n + (975 ) = 0 5n + 05n 950 = 0 n = 05 ± (05) 4 ( 5) ( 950) ( 5) n = 5 or n = 6 Pge 5 of 7 (kshifdeel.com)

6 QUESTION BANK AP (ADVANCED OR WORKING BACKWARDS) 3 The first term of n AP is 4 nd the sum of the first five terms nd the first ten terms re equl in mgnitude but opposite in sign. The 3 rd term of the AP is: () 6 4 (b) 6 (c) 4 (d) None of these ANSWER 3 = 4 s 5 = s 0 [ 5 [ 4 + (5 ) d] Dt Given in question = 0 [ 4 + (0 ) d] [.5(8 + 4d)] = 5(8 + 9d) 70 0d = d 45d + 0d = d = 0 d = 0 55 = 4 nth term = + (n )d 3rd term = 4 + (3 ) 4 = 70 = A person is employed in compny t Rs.3,000 per month nd he would get n increse of Rs. 00 per yer. The totl mount which he receives in 5 yers nd the monthly slry in the lst yers re () Rs.5,400 nd Rs.,50,000 (b) Rs.5,40 nd Rs.,60,000 (c) Rs.5,500 nd Rs.,60,000 (d) Rs.5,400 nd Rs.,60,000 ANSWER 4 n [ + (n ) d] = 36000, d = 00, n = 5 5 [(36000) + (5 ) 00] Monthly figures converted to per nnum,60,000 nth term = + (n )d nth term = (5 )00 = = 5400 Annul slry converted to monthly A quicker wy is to use monthly n Pge 6 of 7 (kshifdeel.com)

7 5 The sum of ll nturl numbers between 500 nd 000 which re divisible by 3 is: () 8,405 (b) 4,805 (c) 8,540 (d) None of these ANSWER 5 n [ + (n ) d] 507 is first term divisible by 3 in given rnge 38 [(507) + (38 ) 3] 38 is n becuse 500/3 is is common difference 6 If unity is dded to the sum of ny number of terms of the A.P. 3, 5,7,9,. The resulting sum () perfect cube (b) perfect squre (c) Both () nd (b) (d) None of these ANSWER = 8 + = 9 = = 5 + = 6 = = 4 + = 5 = 5 Unity () is dded nd perfect squre formed 7 Divide.50 into five prts A.P such tht the first prt nd the lst prt re the rtio of :3 (),.5,.5,.75, 3 (b) -, -.5,-.5, -.75, -3 (c),3.5,5, 6.5,8 (d) 0, 30, 40, 50, 60 ANSWER 7 + ( + d) + ( + d) + ( + 3d) + ( + 4d) = d =.5 + d =.5 (prtilly working bckwrd) Divided by 5 nd it is equl to 3 rd term In ll options this is the vlue of third term in option Pge 7 of 7 (kshifdeel.com)

8 8 The p th term of n A.p is q. The sum of the pqth term is () (pq+) (b) (pq-) (c) Pq+ (d) Pq - ANSWER 8 = + (p q )d = + pd d Eqution q = + (q p )d = + qd d Eqution p q Subtrction side wise = + pd d qd + d p q Simplified = pd qd p p q pq = (p q)d Tking LCM on left side Tking d common on right side d = Dividing both sides by (p-q) pq = + (p ) q pq q = + q pq = pq Sn = pq [ ( ) + (pq ) pq pq ] Sn = pq [ pq + pq pq pq ] Sn = pq [ pq + pq ] Sn = pq [ pq + pq ] Sn = ½ ( + pq ) Sn = ( + pq) Putting vlue of d in eqution LCM Simplified Sum formul for n=pq Vlue of nd d used s clculted bove tken common Pge 8 of 7 (kshifdeel.com)

9 GEOMETRIC PROGRESSION A geometric progression is one where the rtio between term nd the one Definition tht immeditely precedes it is constnt throughout the whole series. The rtio tht links consecutive numbers in the series is known s the Common rtio common rtio. The common rtio cn be found by dividing term into the r next term in the sequence. Sequence Common rtio,, 4, 8, 6, 3, 64, 8 etc.. +, -, 4, -8, 6, -3, 64, -8 etc. -, ½, ¼, /8, /6 etc. ½ Note tht for three consecutive numbers w, x, y nd z in n geometric progression: GP Property nd term 3rd term = st term nd term First term Number of terms n Vlue of nth term Sum of number of terms Sum of n infinite series Solving for n The first term in the series is expressed by constnt The totl number of terms in the series under review is expressed by constnt n Used if the r is or bove nd positive (rn ) r nth term = r n Used if the r is less thn or negtive ( rn ) r If the common rtio is numericlly greter thn unity (i.e. greter thn or equl to or less thn or equl to -) the series will hve n infinite vlue. If the common rtio is between + nd - the series will hve finite vlue. Use log. r Pge 9 of 7 (kshifdeel.com)

10 QUESTION BANK nth term (GP) 9 The 7 th term of the series 6,, 4..is: () 384 (b) 834 (c) 438 (d) None of these ANSWER 9 nth term = r n = 6, r =, n = 7 7th term = 6 7 = The lst term of the series x, x,.. to 3 terms is () x 8 (b) x (c) (d) x 8 None of these ANSWER 0 nth term = r n 3th term = x ( x ) 3 = x, r =, n = 3 x 3th term = x 30 x 30 3th term = x 8 How mny terms re there in the sequence of,,.., 3, 64? () 3 (b) 4 (c) 5 (d) 6 ANSWER nth term = r n GP nd r = 64 = 8 n 89 = n log 89 n = Using log log n = 3 n = 4 Pge 0 of 7 (kshifdeel.com)

11 QUESTION BANK SUM (GP) If you sve pise tody, pise the next dy 4 pise the succeeding dy nd so on, thn your totl svings in two weeks will be: () Rs.63 (b) Rs.83 (c) Rs (d) None of these ANSWER (rn ) r 0.0(4 ) = = 0.0, r =, n = 4 3 The lest vlue of n for which the sum of n terms of the series is greter thn 7,000 is () 9 (b) 0 (c) 8 (d) 7 ANSWER 3 n = (rn ) r 700 = (3n ) = 3 n 4003 = 3 n log 4003 log 3 = 8.69 or 9 =, r = 3, 700 Pge of 7 (kshifdeel.com)

12 QUESTION BANK SUM INFINITY (GP) 4 The sum of the series, 3, 9, 7 to the infinity () (b) 3 (c) 5 5 (d) 5 ANSWER 4 r or 5/ =, r = Sum of the series to infinity is: 8 () 6 (b) 8 (c) 7 (d) 9 ANSWER 5 r = 6, r = The sum of the infinite series is 4 () (b).0000 (c) (d).999 ANSWER 6 r 0.5 =, r = Sum of the infinity of the following geometric progression. (.) () 9 (b) -9 (c) 0 (d) -0 (.) 3 + is: ANSWER 7 r Pge of 7 (kshifdeel.com) = 0.9, r = = 0.9

13 8 The infinite G.P with first term 4 nd the sum 3 is: (),,, (b),, (c),,, (d) None of these ANSWER 8 r = 0.5 r r = 0.5 or 4 therefore 4, 6, 64 = /4, /3 9 Sum of the series, 3, 9, () 3 (c) 3 7. To the infinity is: (b) 3 (d) 3 ANSWER 9 r 3 =, r = /3.5 or 3 Pge 3 of 7 (kshifdeel.com)

14 QUESTION BANK WORKING BACKWARDS (GP) 30 If the sum of infinite terms in G.P is nd the sum of their squres is 4 the series is: 3 (),,. (b),,. 4 4 (c) -,,. (d) None of these 4 ANSWER 30 r Option () Question condition = Condition mtched 4 3 = Option () Question condition Condition mtched 3 The sum of the infinite series () - 4 (b) ( + ) (c) + 4 (d) 4 + ANSWER 3 r =, r = = = ( + ) ( + ) ) + + = 4 + = 4 + Multiplied by common fctor Pge 4 of 7 (kshifdeel.com)

15 QUESTION BANK AP & GP COMBINED 3 Three numbers re in AP nd their sum is, if, 5, 5 re dded to them respectively. They form G.P the numbers re () 5, 7,9 (b) 9, 5, 7 (c) 7, 5,9 (d) None of these ANSWER 3 (working bckwrds is better pproch) 5+7+9= Option () AP Criteri met (5+), (7+5), (9+5)=6,,4 By dding numbers mke it GP nd term 3rd term = Is it vlid GP? st term nd term 6 = 4 GP proved vlid = Option () is nswer 33 The sum of the number in G.P is 70. If the two extremes be multiplied ech by 4 nd the geometric men by 5, the products re in A.P, the numbers re (), 8, 40 (b) 0, 30, 90 (c) 40, 0, 0 (d) None of these ANSWER 33 (working bckwrds is better pproch) = 70 Option C GP criteri is vlid (40x4),(0x5),(0x4)=60,00,40 Forming it in AP ccording to Question nd term st term = 3 rd term nd term Is it vlid AP? = AP proved vlid -60 = -60 Option c is correct Pge 5 of 7 (kshifdeel.com)

16 34 The numbers x, 8, y re in G.P nd the numbers x, y, -8 re in A.P. the vlues of x, y re () 6, 4 (b) 4, 6 (c) Both () nd(b) (d) None of these ANSWER 34 y 8 = 8 x y = 64 x 8 y = y x 8 = y x 8 = 64 x x Three consecutive terms in GP Vlue of y Three consecutive terms in AP Putting y in eqution 8 = 8 x x Multiplied by x 8x = 8 x x 8x 8 = 0 Qudrtic formed x 6x + 8x 8 = 0 x(x 6) + 8(x 6) = 0 (x 6)(x + 8) = 0 x = 6 or x = 8 Vlue of x found by fctoriztion y = 64 x = 64 6 = 4 Putting vlue of x in eqution of y 6, 4 The nswer Working bckwrds would be quicker Pge 6 of 7 (kshifdeel.com)

17 QUESTION BANK Involving finncil mthemtics 35 A person borrows Rs.8, 000 t.76% simple interest per nnum. The principl nd the interest re to be pid in 0 monthly instllments. If ech instllment is double the preceding one, the vlue of the first nd the lst instllments re: () Rs.8 nd Rs (b) Rs.8 nd Rs (c) Rs.7 nd Rs (d) Rs.8 nd Rs ANSWER 35 F = P( + rn) = 8,000 ( ) = 8,84 (rn ) r 8,84 = (0 ) = 8 0th term = r n = 8 0 = 4,096 Future vlue bsed on simple interest To find first term To find 0 th term 36 At 0 % C.I. p., sum of money ccumultes to Rs.9, 65 in 5 yers. The sum invested initilly is: () Rs. 5, (b) Rs. 5,970 (c) Rs. 5,975 (d) Rs. 5, ANSWER 36 P = F ( + r m ) mn P = 965 ( ) 5 P = Credits: Mr. Wseem Seed, Dr. Muneer nd Ms. Anum Iqbl Concept & Compiltion: Kshif Adeel Dted: 4 November 07 Pge 7 of 7 (kshifdeel.com)

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