SEQUENCES AND SERIES

Transcription

1 Sequeces ad 6 Sequeces Ad SEQUENCES AND SERIES Successio of umbers of which oe umber is desigated as the first, other as the secod, aother as the third ad so o gives rise to what is called a sequece. Sequeces have wide applicatios. I this lesso we shall discuss particular types of sequeces called arithmetic sequece, geometric sequece ad also fid arithmetic mea (A.M), geometric mea (G.M) betwee two give umbers. We will also establish the relatio betwee A.M ad G.M. Let us cosider the followig problems : (a) A ma places a pair of ewly bor rabbits ito a warre ad wats to kow how may rabbits he would have over a certai period of time. A pair of rabbits will start producig offsprigs two moths after they were bor ad every followig moth oe ew pair of rabbits will appear. At the begiig the ma will have i his warre oly oe pair of rabbits, durig the secod moth he will have the same pair of rabbits, durig the third moth the umber of pairs of rabbits i the warre will grow to two; durig the fourth moth there will be three pairs of rabbits i the warre. Thus, the umber of pairs of rabbits i the cosecutive moths are :,,,, 5, 8,,... (b) The recurrig decimal 0. ca be writte as a sum 0. = (c) A ma ears Rs.0 o the first day, Rs. 0 o the secod day, Rs. 50 o the third day ad so o. The day to day earig o f the ma may be writte as 0, 0, 50, 70, 90, We may ask what his earigs will be o the 0 th day i a specific moth. Agai let us cosider the followig sequeces: (), 4, 8, 6, (),,,, () 0.0, 0.000, , I these three sequeces, each term except the first, progressess i a defiite order but differet from the order of other three problems. I this lesso we will discuss those sequeces whose term progressess i a defiite order. MATHEMATICS 4

2 Sequeces Ad OBJECTIVES After studyig this lesso, you will be able to : describe the cocept of a sequece (progressio); defie a A.P. ad cite examples; fid commo differece ad geeral term of a A.P; Sequeces ad fid the fourth quatity of a A.P. give ay three of the quatities a, d, ad t ; calculate the commo differece or ay other term of the A.P. give ay two terms of the A.P; derive the formula for the sum of terms of a A.P; calculate the fourth quatity of a A.P. give three of S,, a ad d; isert A.M. betwee two umbers; solve problems of daily life usig cocept of a A.P; state that a geometric progressio is a sequece icreasig or decreasig by a defiite multiple of a o-zero umber other tha oe; idetify G.P. s from a give set of progessios; fid the commo ratio ad geeral term of a G.P; calculate the fourth quatity of a G.P whe ay three of the quatities t, a, r ad are give; calculate the commo ratio ad ay term whe two of the terms of the G.P. are give; write progressio whe the geeral term is give; derive the formula for sum of terms of a G.P; calculate the fourth quatity of a G.P. if ay three of a, r, ad S are give; derive the formula for sum (S ) of ifiite umber of terms of a G.P. whe r ; fid the third quatity whe ay two of S, a ad r are give; covert recurrig decimals to fractios usig G.P; isert G.M. betwee two umbers; ad establish relatioship betwee A.M. ad G.M. EXPECTED BACKGROUND KNOWLEDGE Laws of idices Simultaeous equatios with two ukows. Quadratic Equatios. 4 MATHEMATICS

3 Sequeces ad 6. SEQUENCE A sequece is a collectio of umbers specified i a defiite order by some assiged law, whereby a defiite umber a of the set ca be associated with the correspodig positive iteger. The differet otatios used for a sequece are.. a, a, a,..., a,.... a, =,,,.... {a } Let us cosider the followig sequeces :.,, 4, 8, 6,,...., 4, 9, 6, 5,... Sequeces Ad., 4, 4, 5, 4.,,, 4, 5, 6, I the above examples, the expressio for th term of the sequeces are as give below : () a = () a = () a = for all positive iteger. (4) a = Also for the first problem i the itroductio, the terms ca be obtaied from the relatio a =, a =, a a a, A fiite sequece has a fiite umber of terms. A ifiite sequece cotais a ifiite umber of terms. 6. ARITHMETIC PROGRESSION Let us cosider the followig examples of sequece, of umbers : (), 4, 6, 8, (),,, 5, () 0, 8, 6, 4, (4) 5,,,,, Note that i the above four sequeces of umbers, the first terms are respectively,, 0, ad. The first term has a importat role i this lesso. Also every followig term of the sequece has certai relatio with the first term. What is the relatio of the terms with the first term i Example ()? First term =, Secod term = 4 = + Third term = 6 = + Fourth term = 8 = + ad so o. The cosecutive terms i the above sequece are obtaied by addig to its precedig term. i.e., the differece betwee ay two cosecutive terms is the same. MATHEMATICS 4

4 Sequeces Ad Sequeces ad A fiite sequece of umbers with this property is called a arithmetic progressio. A sequece of umbers with fiite terms i which the differece betwee two cosecutive terms is the same o-zero umber is called the Arithmetic Progressio or simply A. P. The differece betwee two cosecutive terms is called the commo defferece of the A. P. ad is deoted by 'd'. I geeral, a A. P. whose first term is a ad commo differece is d is writte as a, a + d, a + d, a + d, Also we use t to deote the th term of the progressio. 6.. GENERAL TERM OF AN A. P. Let us cosider A. P. a, a + d, a + d, a + d, Here, first term (t ) = a secod term (t ) = a + d = a + ( ) d, third term (t ) = a + d = a + ( ) d By observig the above patter, th term ca be writte as: t = a + ( ) d Hece, if the first term ad the commo differece of a A. P. are kow the ay term of A. P. ca be determied by the above formula. Note.: (i) If the same o-zero umber is added to each term of a A. P. the resultig sequece is agai a A. P. (ii) If each term of a A. P. is multiplied by the same o-zero umber, the resultig sequece is agai a A. P. Example 6. Fid the 0 th term of the A. P.:, 4, 6,... Solutio : Here the first term (a) = ad commo differece d = 4 = Usig the formula t = a + ( ) d, we have t 0 = + (0 ) = + 8 = 0 Hece, the 0th term of the give A. P. is 0. Example 6. The 0 th term of a A. P. is 5 ad st term is 57, fid the 5 th term. Solutio : Let a be the first term ad d be the commo differece of the A. P. The from the formula: t = a + ( ) d, we have t 0 = a + (0 ) d = a + 9d ad t = a + ( ) d = a + 0 d 44 MATHEMATICS

5 Sequeces ad We have, a + 9d = 5...(), a + 0d = 57...() Solve equatios () ad () to get the values of a ad d. Subtractig () from (), we have 4 d = = 4 d Agai from (), a = 5 9d = 5 9 ( ) = = Now t 5 = a + (5 )d = + 4 ( ) = 5 Example 6. Which term of the A. P.: 5,, 7,... is 9? Solutio : Here a = 5, d = 5 = 6 t = 9 We kow that t = a + ( ) d Sequeces Ad 9 = 5 + ( ) 6 ( ) = = 9 = 0 Therefore, 9 is the 0th term of the give A. P. Example 6.4 Is 600 a term of the A. P.:, 9, 6,...? Solutio : Here, a =, ad d = 9 = 7. Let 600 be the th term of the A. P. We have t = + ( ) 7 Accordig to the questio, + ( ) 7 = 600 ( ) 7 = or Sice is a fractio, it caot be a term of the give A. P. Hece, 600 is ot a term of the give A. P. Example 6.5 If a + b + c = 0 ad,, are also i A. P. b c c a a b a b c b c, c a, a b are. i A. P., the prove that Solutio. : Sice a b c b c, c a, a b are i A. P., therefore MATHEMATICS 45

6 Sequeces ad Sequeces Ad or, b a c b c a b c a b c a F b I a c b HG c a K J F I HG b c K J F I HG a b K J F I HG c a K J or, a b c c a a b c a b c b c a b a b c c a or, c a b c a b c a (Sice a + b + c 0) or,,, are i A. P. b c c a a b CHECK YOUR PROGRESS 6.. Fid the th term of each of the followig A. P s. : (a),, 5, 7, (b), 5, 7, 9,. If t = +, the fid the A. P.. Which term of the A. P., 4, 5,... is? Fid also the 0 th term? 4. Is 9 a term of the A. P. 7, 4,,,...? 5. The m th term of a A. P. is ad the th term is m. Show that its (m + ) th term is zero. 6. Three umbers are i A. P. The differece betwee the first ad the last is 8 ad the product of these two is 0. Fid the umbers. 7. The th term of a sequece is a + b. Prove that the sequece is a A. P. with commo differece a. 6. TO FIND THE SUM OF FIRST TERMS IN AN A. P. Let a be the first term ad d be the commo differece of a A. P. Let l deote the last term, i.e., the th term of the A. P. The, l = t = a + ( )d... (i) Let S deote the sum of the first terms of the A. P. The S = a + (a + d) + (a + d) (l d) + (l d) + l... (ii) Reversig the order of terms i the R. H. S. of the above equatio, we have S = l + (l d) + (l d) (a + d) + (a + d) + a... (iii) 46 MATHEMATICS

7 Sequeces ad Addig (ii) ad (iii) vertically, we get S = (a + l) + (a + l) + (a + l) +... cotaiig terms = (a + l) Sequeces Ad i.e., S Also S a l ( ) a d [ ( ) ] [From (i)] It is obvious that t = S S Example 6.6 Fid the sum of terms. Solutio.: Here a =, d = 4 = Usig the formula S a d [ ( ) ], we get S [ ( ) ] ( ) [ ] = ( + ) Example 6.7 The 5 th term of a A. P. is 69. Fid the sum of its 69 terms. Solutio. Let a be the first term ad d be the commo differece of the A. P. We have t 5 = a + (5 ) d = a + 4 d. a + 4 d = (i) Now by the formula, S a d [ ( ) ] We have S a d [ ( 69 ) ] = 69 (a + 4d) [usig (i)] = = 476 Example 6.8 The first term of a A. P. is 0, the last term is 50. If the sum of all the terms is 480, fid the commo differece ad the umber of terms. Solutio : We have: a = 0, l = t = 50, S = 480. MATHEMATICS 47

8 Sequeces Ad By substitutig the values of a, t ad S i the formulae S a d ad t = a+ ( ) d, we get ( ) d (i) 50 = 0 + ( ) d (ii) From (ii), ( ) d = 50 0 = 40 (iii) From (i), we have 480 (0 40) usig (i) or, 60 = From (iii), Sequeces ad 40 8 d (as 6 5) 5 Example 6. 9 Let the th term ad the sum of terms of a A. P. be p ad q respectively. q p Prove that its first term is. Solutio: I this case, t = p ad S = q Let a be the first term of the A. P. or, ( a p ) q Now, S a t q or, a p q or, a q p p a CHECK YOUR PROGRESS 6.. Fid the sum of the followig A. P s. (a) 8,, 4, 7,up to 5 terms (b) 8,,, 7,,up to terms.. How may terms of the A. P.: 7,, 9, 5,... have a sum 95? 48 MATHEMATICS

9 Sequeces ad. A ma takes a iterest-free loa of Rs. 740 from his fried agreeig to repay i mothly istalmets. He gives Rs. 00 i the first moth ad dimiishes his mothly istalmets by Rs. 0 each moth. How may moths will it take to repay the loa? 4. How may terms of the progressio, 6, 9,, must be take at the least to have a sum ot less tha 000? 5. I a childre potato race, potatoes are placed metre apart i a straight lie. A competitor starts from a poit i the lie which is 5 metre from the earest potato. Fid a expressio for the total distace ru i collectig the potatoes, oe at a time ad brigig them back oe at a time to the startig poit. Calculate the value of if the total distace ru is 6 metres. 6. If the sum of first terms of a sequece be a + b, prove that the sequece is a A. P. ad fid its commo differece? Sequeces Ad 6.4 ARITHMETIC MEAN (A. M.) Whe three umbers a, A ad b are i A. P., the A is called the arithmetic mea of umbers a ad b. We have, A a = b A or, A = a b Thus, the required A. M. of two umbers a ad b is a b. Cosider the followig A. P :, 8,, 8,, 8,. There are five terms betwee the first term ad the last term. These terms are called arithmetic meas betwee ad. Cosider aother A. P. :,,,. I this case there are two arithmetic meas, ad betwee ad. Geerally ay umber of arithmetic meas ca be iserted betwee ay two umbers a ad b. Let A, A, A,..., A be arithmetic meas betwee a ad b, the. a, A, A, A..., A, b is a A. P. Let d be the commo differece of this A. P. Clearly it cotais ( + ) terms b = ( + ) th term = a + ( + ) d b a d F HG b a Now, A = a d A a I K J...(i) MATHEMATICS 49

10 Sequeces Ad F HG ( b a) A = a d A a I K J...(ii) Sequeces ad F HG ( b a) A = a d A a These are required arithmetic meas betwee a ad b. Addig (i), (ii),..., (), we get A + A A = a I K J b a... () ( ) ( ) ( ) a b a a b a a b = [Sigle A. M. betwee a ad b] Example 6.0 Isert five arithmetic meas betwee 8 ad 6. Solutio : Let A, A, A, A 4 ad A 5 be five arithmetic meas betwee 8 ad 6. Therefore, 8, A, A, A, A 4, A 5, 6 are i A. P. with a = 8, b = 6, = 7 We have 6 = 8 + (7 ) d d = A = a + d = 8 + =, A = a + d = 8 + = 4 A = a + d = 7, A 4 = a + 4d = 0, A 5 = a + 5d = Hece, the five arithmetic meas betwee 8 ad 6 are, 4, 7, 0 ad. Example 6. The '', A. M's betwee 0 ad 80 are such that the ratio of the first mea ad the last mea is :. Fid the value of. Solutio : Here, 80 is the (+) th term of the A. P., whose first term is 0. Let d be the commo differece. 80 = 0 + (+ ) d or, 80 0 = (+) d or, d = The first A. M. = 0 = = The last A. M. = 0 = MATHEMATICS

11 Sequeces ad We have : = : or, or, 4 + = + or, = 4 4 The umber of A. M's betwee 0 ad 80 is. CHECK YOUR PROGRESS 6. Sequeces Ad. Prove that if the umber of terms of a A. P. is odd the the middle term is the A. M. betwee the first ad last terms.. Betwee 7 ad 85, m umber of arithmetic meas are iserted so that the ratio of (m ) th ad m th meas is : 4. Fid the value of m.. Prove that the sum of arithmetic meas betwee two umbers is times the sigle A. M. betwee them. 4. If the A. M. betwee p th ad q th terms of a A. P., be equal ad to the A. M. betwee r th ad s th terms of the A. P., the show that p + q = r + s. 6.5 GEOMETRIC PROGRESSION Let us cosider the followig sequece of umbers : (),, 4, 8, 6, (),,, 9, 7 (),, 9, 7, (4) x, x, x, x 4, If we see the patters of the terms of every sequece i the above examples each term is related to the leadig term by a defiite rule. For Example (), the first term is, the secod term is twice the first term, the third term is times of the leadig term. Agai for Example (), the first term is, the secod term is times of the first term, third term is times of the first term. A sequece with this property is called a gemetric progressio. A sequece of umbers i which the ratio of ay term to the term which immediately precedes is the same o zero umber (other tha), is called a geometric progressio or simply G. P. This ratio is called the commo ratio. Thus, Secod term Third term First term Secod term... is called the commo ratio of the geometric progressio. MATHEMATICS 5

12 Sequeces ad Sequeces Ad Examples () to (4) are geometric progressios with the first term,,, x ad with commo ratio,,, ad x respectively. The most geeral form of a G. P. with the first term a ad commo ratio r is a, ar, ar, ar, GENERAL TERM Let us cosider a geometric progressio with the first term a ad commo ratio r. The its terms are give by a, ar, ar, ar,... I this case, t = a = ar - t = ar = ar t = ar = ar t 4 = ar = ar O geeralisatio, we get the expressio for the th term as t = ar... (A) 6.5. SOME PROPERTIES OF G. P. (i) If all the terms of a G. P. are multiplied by the same o-zero quatity, the resultig series is also i G. P. The resultig G. P. has the same commo ratio as the origial oe. If a, b, c, d,... are i G. P. the ak, bk, ck, dk... are also i G. P. k 0 (ii) If all the terms of a G. P. are raised to the same power, the resultig series is also i G. P. Let a, b, c, d... are i G. P. the a k, b k, c k, d k,... are also i G. P. k 0 The commo ratio of the resultig G. P. will be obtaied by raisig the same power to the origial commo ratio. Example 6. Fid the 6 th term of the G. P.: 4, 8, 6,... Solutio : I this case the first term (a) = 4 Commo ratio (r) = 8 4 = Now usig the formula t = ar, we get t 6 = 4 6 = 4 = 8 Hece, the 6 th term of the G. P. is 8. Example 6. The 4 th ad the 9 th term of a G. P. are 8 ad 56 respectively. Fid the G. P. Solutio : Let a be the first term ad r be the commo ratio of the G. P., the t 4 = ar 4 = ar, t 9 = ar 9 = ar 8 Accordig to the questio, ar 8 = 56 () ad ar = 8 () 5 MATHEMATICS

13 Sequeces ad ar ar 8 56 or r 5 = = 5 r = 8 Sequeces Ad 8 Agai from (), a = 8 a 8 Therefore, the G. P. is,, 4, 8, 6,... Example 6.4 Which term of the G. P.: 5, 0, 0, 40,... is 0? Solutio : I this case, a = 5; r 0 5. Suppose that 0 is the th term of the G. P. By the formula,t = ar, we get t = 5. ( ) 5. ( ) = 0 (Give) ( ) = 64 = ( ) 6 = 6 = 7 Hece, 0 is the 7 th term of the G. P. Example 6.5 If a, b, c, ad d are i G. P., the show that (a + b), (b + c), ad (c + d) are also i G. P. Solutio. Sice a, b, c, ad d are i G. P., b c a b b = ac, c = bd, ad = bc...() d c Now, (a + b) (c + d) a bc d = (ac + bc + ad + bd) = (b + c + bc)...[usig ()] = ( b c) ( c d ) ( b c) ( b c) Thus, (a + b) ( a b), (b + c), (c + d) are i G. P. CHECK YOUR PROGRESS 6.4. The first term ad the commo ratio of a G. P. are respectively ad the first five terms.. Write dow MATHEMATICS 5

14 Sequeces Ad Sequeces ad. Which term of the G. P.,, 4, 8, 6,... is 04? Is 50 a term of the G. P.?. Three umbers are i G. P. Their sum is 4 ad their product is 6. Fid the umbers i proper order. 4. The th term of a G. P. is for all. Fid (a) the first term (b) the commo ratio of the G. P. 6.6 SUM OF TERMS OF A G. P. Let a deote the first term ad r the commo ratio of a G. P. Let S represet the sum of first terms of the G. P. Thus, S = a + ar + ar ar + ar... () Multiplyig () by r, we get r S = ar + ar ar + ar + ar... () () () S rs = a ar or S ( r) = a ( r ) S a r ( ) r...(a) a r ( ) r...(b) Either (A) or (B) gives the sum up to the th term whe r. It is coveiet to use formula (A) whe r < ad (B) whe r >. Example 6.6 Fid the sum of the G. P.:,, 9, 7,... up to the 0 th term. Solutio : Here the first term (a) = ad the commo ratio Now usig the formula, S r 0 0 a r.( ), ( r >) we get S 0 r Example 6.7 Fid the sum of the G. P.:,,,,, 8 Solutio : Here, a ; r ad t = l = 8 Now t = 8 ( ) ( ) 4 8 ( ) ( ) = 8 or = 0 54 MATHEMATICS

15 Sequeces ad S 0 = 0 Sequeces Ad Example 6.8 Fid the sum of the G. P.: 0.6, 0.06, 0.006, , to terms. Solutio. Here, a = 0.6 = 6 0 ad r Usig the formula S a r ( ), we have [ r <] r S 0 I HG K J. Hece, the required sum is 0 F Example 6.9 How may terms of the G. P.: 64,, 6, has the sum 7? Solutio : Here, a 64, r 64 (<) ad S 7 Usig the formula S S R S T 64 a r ( ), we get r F HG I K J U V W F HG I K J U V R S T W (give) or or 55 8 or F F H G I K J I HG K J F H G I K J = 8 Thus, the required umber of terms is 8. MATHEMATICS 55

16 Sequeces Ad Sequeces ad Example 6.0 Fid the sum of the followig sequece :,,,... to terms. Solutio : Let S deote the sum. The S = to terms = ( to terms) = 9 ( to terms) to terms ) to terms to terms R S T L N M ( 0 ) U VW O QP [ is a G P with r = 0<] Example 6. Fid the sum up to terms of the sequece: 0.7, 0.77, 0.777, Solutio : Let S deote the sum, the S = to terms = 7( to terms) = 7 9 ( to terms) = 7 9 ( 0.) + ( 0.0) + ( 0.00) + to terms = 7 9 ( terms) ( to terms) 7 to terms MATHEMATICS

17 Sequeces ad R S F HG I K J U V W (Sice r < ) = T 0 = L NM O QP = Sequeces Ad CHECK YOUR PROGRESS 6.5. Fid the sum of each of the followig G. P's : (a) 6,, 4,... to 0 terms (b),, 4, 8,... to 0 terms. 6. How may terms of the G. P. 8, 6,, 64, have their sum 884?. Show that the sum of the G. P. a + b l is bl a b a 4. Fid the sum of each of the followig sequeces up to terms. (a) 8, 88, 888,... (b) 0., 0., 0., INFINITE GEOMETRIC PROGRESSION So far, we have foud the sum of a fiite umber of terms of a G. P. We will ow lear to fid out the sum of ifiitely may terms of a G P such as.,, 4, 8, 6, We will proceed as follows: Here a, r. The th term of the G. P. is t = ad sum to terms i.e., S. So, o matter, how large may be, the sum of terms is ever more tha. So, if we take the sum of all the ifiitely may terms, we shall ot get more tha as aswer. Also ote that the recurrig decimal 0. is really i.e., 0. is actually the sum of the above ifiite sequece. MATHEMATICS 57

18 Sequeces Ad Sequeces ad O the other had it is at oce obvious that if we sum ifiitely may terms of the G. P.,, 4, 8, 6,... we shall get a ifiiite sum. So, sometimes we may be able to add the ifiitely may terms of G. P. ad sometimes we may ot. We shall discuss this questio ow SUM OF INFINITE TERMS OF A G. P. Let us cosider a G. P. with ifiite umber of terms ad commo ratio r. Case : We assume that r > The expressio for the sum of terms of the G. P. is the give by S a( r ) a r a r r r... (A) Now as becomes larger ad larger r also becomes larger ad larger. Thus, whe is ifiitely large ad r > the the sum is also ifiitely large which has o importace i Mathematics. We ow cosider the other possibility. Case : Let r < a ( r ) a ar Formula (A) ca be writte as S r r r Now as becomes ifiitely large, r becomes ifiitely small, i.e., as, r 0, the the above expressio for sum takes the form S a r Hece, the sum of a ifiite G. P. with the first term a ad commo ratio r is give by a S r, whe r <...(i) Example 6. Fid the sum of the ifiite G. P.,, 4, 8, Solutio : Here, the first term of the ifiite G. P. is a, ad r 9. Here, r 58 MATHEMATICS

19 Sequeces ad Usig the formula for sum S a r Hece, the sum of the give G. P. is 5. we have S F H G I K J 5 Sequeces Ad Example 6. Express the recurrig decimal 0. as a ifiite G. P. ad fid its value i ratioal form. Solutio. 0. = = The above is a ifiite G. P. with the first term a 0 ad 0 r 0 0 Hece, by usig the formula S a, r we get Hece, the recurrig decimal 0. =. Example 6.4 The distace travelled (i cm) by a simple pedulum i cosecutive secods are 6,, 9,... How much distace will it travel before comig to rest? Solutio : The distace travelled by the pedulum i cosecutive secods are, 6,, 9,... is a ifiite geometric progressio with the first term a = 6 ad r. 6 4 Hece, usig the formula S a r we have 6 6 S 64 Distace travelled by the pedulum is 64 cm. 4 4 MATHEMATICS 59

20 Sequeces Ad Sequeces ad Example 6.5 The sum of a ifiite G. P. is ad sum of its first two terms is 8. Fid the first term. Solutio: I this problem S =. Let a be the first term ad r be the commo ratio of the give ifiite G. P. 8 The accordig to the questio. a ar or, a( r) 8... () Also from S a, a r we have r or, a ( r)... () From () ad (), we get.. ( r) ( + r) = 8 8 or, r or, r 9 9 or, r From (), a = = or 4 accordig as r. CHECK YOUR PROGRESS 6.6 () Fid the sum of each of the followig iifiite G. P's : (a) (b) Express the followig recurrig decimals as a ifiite G. P. ad the fid out their values as a ratioal umber. (a) 0.7 (b) 0.5. The sum of a ifiite G. P. is 5 ad the sum of the squares of the terms is 45. Fid the G.P. 4. The sum of a ifiite G. P. is ad the first term is. Fid the G.P MATHEMATICS

21 Sequeces ad 6.8 GEOMETRIC MEAN (G. M.) If a, G, b are i G. P., the G is called the geometric mea betwee a ad b. If three umbers are i G. P., the middle oe is called the geometric mea betwee the other two. If a, G, G,..., G, b are i G. P., the G, G,... G are called G. M.'s betwee a ad b. The geometric mea of umbers is defied as the th root of their product. Thus if a, a,..., a are umbers, the their Sequeces Ad G. M. = (a, a,... a ) Let G be the G. M. betwee a ad b, the a, G, b are i G. P G b a G or, G = ab or, G = ab Geometric mea = Product of extremes Give ay two positive umbers a ad b, ay umber of geometric meas ca be iserted betwee them Let a, a, a..., a be geometric meas betwee a ad b. The a, a, a,... a, b is a G. P. Thus, b beig the ( + ) th term, we have b = a r + or, b r or, a b r a Hece, a = ar = a F H G I K J b a, a = ar = b a a b a a ar a Further we ca show that the product of these G. M.'s is equal to th power of the sigle geometric mea betwee a ad b. Multiplyig a. a,... a, we have MATHEMATICS 6

22 Sequeces Ad b a, a a a a = a F bi HG a K J = ( ab) b a a Sequeces ad ( ) ( ) b a a d abi G = (sigle G. M. betwee a ad b) Example 6.6 Fid the G. M. betwee ad 7 Solutio : We kow that if a is the G. M. betwee a ad b, the G ab G. M. betwee ad Example 6.7 Isert three geometric meas betwee ad 56. Solutio : Let G, G, G, be the three geometric meas betwee ad 56. The, G, G, G, 56 are i G. P. If r be the commo ratio, the t 5 = 56 i.e, ar 4 = 56. r 4 = 56 or, r = 6 or, r = 4 Whe r = 4, G =. 4 = 4, G =. (4) = 6 ad G =. (4) = 64 Whe r = 4, G = 4, G = () ( 4) = 6 ad G = () ( 4) = 64 G.M. betwee ad 56 are 4, 6, 64, or, 4, 6, 64. Example 6.8 If 4, 6, 4 are i G. P. isert two more umbers i this progressio so that it agai forms a G. P. Solutio : G. M. betwee 4 ad 6 = G. M. betwee 6 ad If we itroduce betwee 4 ad 6 ad 08 betwe 6 ad 4, the umbers 4,, 6, 08, 4 form a G. P. The two ew umbers iserted are ad Example 6.9 Fid the value of such that a a a ad b. b b may be the geometric mea betwee MATHEMATICS

23 Sequeces ad Solutio : If x be G. M. betwee a ad b, the x a b a a b b a or, a b a b a b F HG b I KJ or, a b a b a b or, a a b b a b F H G or, a a. b a b b or, a b I KJ Sequeces Ad or, a b or, F ai HG b K J a b F H G I K J 0 = 0 or, 6.8. RELATIONSHIP BETWEEN A. M. AND G.M. Let a ad b be the two umbers. Let A ad G be the A. M. ad G. M. respectively betwee a ad b A = a b ab, G A G = a b ab = d i d i a b ab = d a bi 0 A > G Example 6.0 The arithemetic mea betwee two umbers is 4 ad their geometric mea is 6. Fid the umbers. Solutio : Let the umbers be a ad b. Sice A. M. betwee a ad b is 4, a b = 4, or, a + b = () Sice G. M. betwee a ad b is 6, ab = 6 or, ab = 56 we kow that (a b) = (a + b) 4 ab () = (68) 4 56 = = 600 MATHEMATICS 6

24 Sequeces Ad Sequeces ad a b = 600 = 60 () Addig () ad (), we get, a = 8 a = 64 Subtractig () from (), we get b = 8 or, b = 4 Required umbers are 64 ad 4. Example 6. The arithmetic mea betwee two quatities b ad c is a ad the two geometric meas betwee them are g ad g. Prove that g + g = abc Solutio : The A. M. betwee b ad c is a b c = a, or, b + c = a Agai g ad g are two G. M.'s betwee b ad c b, g, g, c are i G. P. If r be the commo ratio, the c = br ci or, r = HG b K J F g = br = b c I HG b K J ad g = br = LF NM g + g = b c c MHG I bk J F H G I bk J O P Q P = b c b b c b F HG F c b = bc (a) [sice b + c = a] = abc I K J = b c F HG b c b I K J Example 6. The product of first three terms of a G. P. is 000. If we add 6 to its secod term ad 7 to its rd term, the three terms form a A. P. Fid the terms of the G. P. a Solutio : Let t, t a ad t r ar be the first three terms of G. P. The, their product = a r. a. ar = 000 or, a = 000, or, a = 0 By the questio, t, t + 6, t + 7 are i A. P....() 64 MATHEMATICS

25 Sequeces ad i.e. a, a + 6, ar + 7 are i A. P. r (a + 6) a r = (ar + 7) (a + 6) or, a ( a 6) ( ar 7 ) r 0 or, ( 0 6) ( 0r 7) [usig ()] r or, r = r + 7r or, 0r 5r + 0 = 0 Sequeces Ad r , Whe a = 0, r =. the the terms are 0, 0() i.e., 5, 0, 0 Whe a = 0, r the the terms are 0(), 0, 0 CHECK YOUR PROGRESS 6.7. Isert 8 G. M.'s betwee 8 ad 64. F I HG K J i.e., 0, 0, 5. If a is the first of geometric meas betwee a ad b, show that a + = a b. If G is the G. M. betwee a ad b, prove that G a G b G 4. If the A. M. ad G. M. betwee two umbers are i the ratio m :, the prove that the umbers are i the ratio m m : m m 5. If A ad G are respectvely arithmetic ad geometric meas betwee two umbers a ad b, the show that A > G. 6. The sum of first three terms of a G. P. is ad their product is. Fid the G. P. 7. The product of three terms of a G. P. is 5. If 8 is added to first ad 6 is added to secod term, the umbers form a A. P., Fid the umbers. A C % + LET US SUM UP A sequece i which the differece of two cousecutive terms is always costat ( 0) is called a Arithmetic Progressio (A. P.) MATHEMATICS 65

26 Sequeces Ad Sequeces ad The geeral term of a A. P. a, a + d, a + d,... is give by t = a + ( ) d S, the sum of the first terms of the A.P a, a+d, a+d,... is give by S a d = (a + l), where l = a + ( ) d. t = S S A arithmetic mea betwee a ad b is a b A sequece i which the ratio of two cosecutive terms is always costat ( 0) is called a Geometric Progressio (G. P.) The th term of a G. P.: a, ar, ar,... is ar Sum of the first terms of a G. P.: a, ar, ar,... is. S = a r ( ) r for r > = a r ( ) for r < r The sums of a ifitite G. P. a, ar, ar,... is give by a S = r for r < Geometric mea G betwee two umbers a ad b is ab The arithmetic mea A betwee two umbers a ad b is always greater tha the correspodig Geometric mea G i.e., A > G. SUPPORTIVE WEB SITES MATHEMATICS

27 Sequeces ad TERMINAL EXERCISE. Fid the sum of all the atural umbers betwee 00 ad 00 which are divisible by 7.. The sum of the first terms of two A. P.'s are i the ratio ( ) : ( + ). Fid the ratio of their 0 th terms.. If a, b, c are i A. P. the show that b + c, c + a, a + b are also i A. P. 4. If a, a,..., a are i A. P., the prove that Sequeces Ad... a a a a a a a a a a 4 5. If (b c), (c a), (a b) are i A. P., the prove that b c, c a, a b, are also i A. P. 6. If the p th, q th ad r th terms are P, Q, R respectively. Prove that P (Q R) + Q (R P) + r (P Q) = 0. F I HG K J 7. If a, b, c are i G. P. the prove that a b c a b c 8. If a, b, c, d are i G. P., show that each of the followig form a G. P. : a b c (a) (a b ), (b c ), (c d ) (b),, a b b c c d 9. If x, y, z are the p th, q th ad r th terms of a G. P., prove that x q r y r p z p q = 0. If a, b, c are i A. P. ad x, y, z are i G. P. the prove that x b c y c a z a b =. If the sum of the first terms of a G. P. is represeted by S, the prove that S (S S ) = (S S ). If p, q, r are i A. P. the prove that the p th, q th ad r th terms of a G. P. are also i G. P.. If S =... + S 00, fid the least value of such that 4. If the sum of the first terms of a G. P. is S ad the product of these terms is p ad the sum of their reciprocals is R, the prove that p MATHEMATICS 67 S R F H G I K J

28 Sequeces Ad Sequeces ad ANSWERS CHECK YOUR PROGRESS 6.. (a) (b) +., 5, 7, 9,.... 0, 6 4. o 5. m , 6,, CHECK YOUR PROGRESS 6.. (a) 45 (b) , 9 6. a CHECK YOUR PROGRESS CHECK YOUR PROGRESS 6. 4.,, 4, 8, 6. th, o. 6, 6, or, 6, 6 4. (a) 6 (b) CHECK YOUR PROGRESS 6.5. (a) 68 (b) 4. (a) F I HG K J c h (b) CHECK YOUR PROGRESS F HG 8 0 I K J. (a) (b) 4. (a) 7 9 (b) , , 9, 7, ,,,, CHECK YOUR PROGRESS ,,,,,,,, TERMINAL EXERCISE : ,, 4... or,, , 8, MATHEMATICS

29 Some Special Sequeces 7 Sequeces ad SOME SPECIAL SEQUENCES Suppose you are asked to collect pebbles every day i such a way that o the first day if you collect oe pebble, secod day you collect double of the pebbles that you have collected o the first day, third day you collect double of the pebbles that you have collected o the secod day, ad so o. The you write the umber of pebbles collected daywise, you will have a sequece,,,,,... From a sequece we derive a series. The series correspodig to the above sequece is Oe well kow series is Fiboacci series I this lesso we shall study some special types of series i detail. OBJECTIVES After studyig this lesso, you will be able to : defie a series; calculate the terms of a series for give values of from t ; evaluate,, usig method of differeces ad mathematical iductio; ad evaluate simple series like terms. EXPECTED BACKGROUND KNOWLEDGE Cocept of a sequece Cocept of A. P. ad G. P., sum of terms. Kowldge of covertig recurrig decimals to fractios by usig G. P. 7. SERIES A expressio of the form u + u + u u +... is called a series, where u, u,u..., u... is a sequece of umbers. The above series is deoted by MATHEMATICS 69 r u r. If is fiite

30 Sequeces ad Some Special Sequeces the the series is a fiite series, otherwise the series is ifiite. Thus we fid that a series is associated to a sequece. Thus a series is a sum of terms arraged i order, accordig to some defiite law. Cosider the followig sets of umbers : (a), 6,,..., (b),,, 6 9 (c) 48, 4,,..., (d),,,... (a), (b), (c), (d) form sequeces, sice they are coected by a defiite law. The series associated with them are : , , , Example 7. Write the first 6 terms of each of the followig sequeces, whose th term is give by (a) T = +, (b) a = + (c) f = ( ). 5 Hece fid the series associated to each of the above sequeces. Solutio : (a) T = +, For =, T =. + =, For =, T =. + = 5 For =, T =. + = 7, For = 4, T 4 =.4 + = 9 For = 5, T 5 =.5 + =, For = 6, T 6 =.6 + = Hece the series associated to the above sequece is (b) a = +, For =, a = + = For =, a = + =, For =, a = + = 7 For = 4, a 4 = =,For = 5, a 5 = = For = 6, a 6 = = Hece the series associated to the above sequece is (c) Here f = ( ) 5, For =, f = ( ) 5 = 5 For =, f = ( ) 5 = 5, For =, f = ( ) 5 = 5 For = 4, f 4 = ( ) = 65, For = 5, f 5 = ( ) = 5 For = 6, f 6 = ( ) = 565 The correspodig series relative to the sequece f = ( ) 5 is MATHEMATICS

31 Some Special Sequeces Example 7. Write the th term of each of the followig series : (a) (b) (c) (d) Solutio : (a) The series is Here the odd terms are egative ad the eve terms are positive. The above series is obtaied by multiplyig the series by T = ( ) = ( ) (b) The series is Sequeces ad T = ( ) + (c) The series is The above series ca be write as i.e., th term, T = 4. (d) The series is i.e., th term is T =. CHECK YOUR PROGRESS 7.. Write the first 6 terms of each of the followig series, whose th term is give by (a) T ( ) ( ) 6 (b) a. If A = ad A =, fid A 6 if A =, A A. Write the th term of each of the followig series: (a) (b) SUM OF THE POWERS OF THE FIRST NATURAL NUMBERS (a) The series of first atural umbers is Let S = This is a arithmetic series whose the first term is, the commo differece is ad the umber MATHEMATICS 7

32 Sequeces ad of terms is. S. i.e., (b) S We ca write Some Special Sequeces Determie the sum of the squares of the first atural umbers. Let S = Cosider the idetity : ( ) = + By givig the values for =,,,...,, i the above idetity, we have. 0 =.. + =.. + =.. + ( ) = + Addig these we get 0 = ( ) ( ) + ( times) or, = S L NM O L NM ( ) ( )... QP or, S ( ) = ( ) ( ) F HG I K J = ( ) = ( ) ( ) S ( ) ( ) e 6 O QP ( ) ( ) 6 (c) Determie the sum of the cubes of the first atural umbers. Here S = Cosider the idetity : 4 ( ) 4 = MATHEMATICS

33 Some Special Sequeces Puttig successively,,,... for we have = = = ( ) 4 = Addig these, we get = 4( ) 6( ) + 4 ( ) ( times) 4 4. S 6 L NM ( ) ( ) 6 O QP 4 4S = 4 + ( + ) ( + ) ( + ) + = 4 + ( + + ) + = = = ( + + ) Sequeces ad i.e., 4S = ( + ) S R S T ( ) ( ) 4 U VW ( ) or, ( ) Note : I problems o fidig sum of the series, we shall fid the th term of the series (t ) ad the use S = t. Example 7. Fid the sum of first terms of the series Solutio : Let S = The th term of the series t = { th term of,, 5,...} { th term of, 5, 7,...} = ( ) ( + ) = 4 MATHEMATICS 7

34 Sequeces ad S = t = 4 ( ) ( ) = 4 ( ) = 4 6 Some Special Sequeces = ( ) ( ) = ( ) = 4 6 Example 7.4 Fid the sum of first terms of the series Solutio : Here t = { + ( )} = ( + ) =. ( + + ) i.e., t = + + Let S = ( + ). = R S T S = t = ( + + ) = + +. U VW L NM O QP ( ) ( ) ( ) ( ) 6 = ( ) L NM ( ) 4 O QP = ( ) ( 0 ) = ( ) ( ) ( 5) Example 7.5 Fid the sum of first terms of the series Solutio : Let S = th term of the series t = { th term of,, 4,...} { th term of, 5, 7,...} { th term of 5, 7, 9,...} = ( + ) ( + ) ( + ) = ( + ) [ ] = S = t = [ ] = () 74 MATHEMATICS

35 Some Special Sequeces ( ) ( ) ( ) ( ) = Sequeces ad ( ) = ( ) ( ) ( ) = ( ) 4 ( ) ( ) ( ) 6 = ( ) 4 ( ) 7 5 Example 7.6 Fid the sum of first terms of the followig series : Solutio : t ( ) ( ) F HG Now puttig successively for =,,,... L N M t t t t L N M L N M 5 L NM O QP O 5QP O 7QP I KJ ( ) ( ) Addig, t t t ( ) O QP MATHEMATICS 75

36 Sequeces ad CHECK YOUR PROGRESS 7. Some Special Sequeces. Fid the sum of first terms of each of the followig series : (a) + ( + ) + ( + + 5) +... (b) (c) () + ( + ) + ( + + ) + ( ) Fid the sum of terms of the series. whose th term is ( + ) ( + 4). Fid the sum of the series upto terms A C % + LET US SUM UP A expressio of the form u + u + u u +... is called a series, where u, u, u... u,... is a sequece of umbers r r ( ) r r r r ( ) ( ) 6 R S T ( ) U VW S = t SUPPORTIVE WEB SITES TERMINAL EXERCISE. Fid the sum of each of the followig series : (a) up to 40 terms. 76 MATHEMATICS

37 Some Special Sequeces (b) up to 6 terms.. Sum each of the followig series to terms : (a) Sequeces ad (b) (c) Fid the sum of first terms of the series Fid the sum to terms of the series Fid the sum to terms of the series Fid the sum of () 7. Show that... ( )... ( ) 5 MATHEMATICS 77

38 Sequeces ad ANSWERS CHECK YOUR PROGRESS 7. Some Special Sequeces. (a), 4, 0, 0, 5, 56 (b) ,,,,, (a) ( ) (b) ( ) + CHECK YOUR PROGRESS 7.. (a) 6 ( ) ( ) (b) (c) ( ) 4. ( ) 4. ( )( )( ) 4 TERMINAL EXERCISE. (a) 640 (b) 78. (a) + (b) 4 ( ) ( ) (c) ( ) ( ) ( 5 ) F H G I K J 6. ( 8 ) ( ) ( ) 78 MATHEMATICS