SEQUENCES AND SERIES


 Douglas Barrie Robinson
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1 9 SEQUENCES AND SERIES INTRODUCTION Sequeces have may importat applicatios i several spheres of huma activities Whe a collectio of objects is arraged i a defiite order such that it has a idetified first member, secod member, third member ad so o, we say that the collectio is listed i a sequece For example : (i) The amout of moey i a fixed deposit i a bak over a umber of years occur i a sequece (ii) Depreciated values of certai commodities occur i a sequece (iii) The populatio of bacteria at differet times forms a sequece (iv) Cosider the umber of acestors ie parets, gradparets, great gradparets etc that a perso has over 0 geeratios The umbers of perso s acestors for the first, secod, third,, teth geeratios are,, 8,, 0 These umbers form a sequece Sequeces, followig specific patters are called progressios I the previous class, you have already studied about arithmetic progressio (AP) I this chapter, besides studyig more about AP, we shall also study arithmetic mea (AM), geometric progressio (GP), geometric mea (GM), relatioship betwee AM ad GM, arithmeticogeometric series ad sum to terms of special series Σ, Σ, Σ etc 9 SEQUENCE A set of umbers arraged i a defiite order accordig to some defiite rule (or rules) is called a sequece Each umber of the set is called a term of the sequece A sequece is called fiite or ifiite accordig as the umber of terms i it is fiite or ifiite The differet terms of a sequece are usually deoted by a, a, or by T, T, T, The subscript (always a atural umber) deotes the positio of the term i the sequece The umber occurrig at the th place of a sequece ie a is called the geeral term of the sequece A fiite sequece is described by a, a,, a ad a ifiite sequece is described by a, a, to If all the terms are real, we have a real sequece; if all terms are complex umbers, we have a complex sequece etc For example, cosider the followig sequeces : (i), 5, 7, 9,, (ii) 8, 5,,,, (iii), 6, 8, 5,, 58 (iv), 8 (v),, 9, 6, (vi),, 5, 7,,, (vii),,,, 5, 8,,
2 98 MATHEMATICS XI We observe the followig : (i) Here each term is obtaied by addig to the previous term (ii) Here each term is obtaied by subtractig from the precedig term (iii) Here each term is obtaied by multiplyig the precedig term by (iv) Here each term is obtaied by multiplyig the precedig term by (v) Here each term is obtaied by squarig the ext atural umber (vi) This is the sequece of prime umbers (vii) Here each term after secod term is obtaied by addig the previous two terms Also ote that sequeces (i) ad (iii) are fiite sequeces whereas others are ifiite sequeces Moreover to defie a sequece, we eed ot always have a explicit formula for the th term Till today, obody has foud the formula for th prime umber Also ote that, i (i) a = a + (ii) a = a (iii) a = a (iv) a = a (v) a = (vii) a = a + a ( > ) ad i (vi) we may describe a = th prime umber If the terms of a sequece ca be described by a explicit formula, the the sequece is called a progressio Note that the sequeces (i) to (v) give above are all progressios, whereas sequece (vi) is ot a progressio The sequece (vii) ie,,,, 5, 8,,, is also a progressio It is called Fiboacci sequece 9 SERIES If the terms of a sequece are coected by plus sigs we get a series Thus, if a, a,, is a give sequece the the expressio a + a + is called the series associated with the give sequece The series is fiite or ifiite accordig as the give sequece is fiite or ifiite From sequeces (i) to (v) give above, we ca form followig series : (i) (ii) ( ) + ( ) + (iii) (iv) (v) If a deotes the geeral term of a sequece, the a + a + + a is a series of terms I a series a + a + + a k +, the sum of first terms is deoted by S Thus S = a + a + + a = a k k = If S deotes the sum of terms of a sequece, the S S = (a + a + + a ) (a + a + + a ) = a Thus, a = S S REMARK The word series is referred to the idicated sum ot to the sum itself For example, is a fiite series with five terms By the words sum of a series will mea the umber that results from addig the terms, so the sum of the above series is 5
3 SEQUENCES AND SERIES 99 ILLUSTRATIVE EXAMPLES Example Fid the ext term of the sequece (i),, 6, 8 (ii),,, (iii), 8,, 8 (iv),, 5, 7 (v), 8, 7, 6 Solutio (i) We see that each term is obtaied by addig to the previous term Hece, the ext term = 8 + = 0 (ii) Here we see that each term is obtaied by multiplyig the previous term by Hece, the ext term = = 8 (iii) We see that each term is obtaied by multiplyig the previous term by Hece, the ext term = 8 = 5 (iv) Here each term is obtaied by subtractig from the previous term Hece, the ext term = 7 = 9 (v) We see that terms are cubes of atural umbers,,, Hece, the ext term of the sequece is 5 ie 5 Example Write the first four terms of the sequece defied by (i) a = + (ii) a = th prime umber Solutio (i) Give a = + Puttig =,,,, we get a = + = 7, a = + = 9, a = + = 9, a = + = 67 Hece, the first four terms of the give sequece are 7, 9, 9, 67 (ii) We kow that the first four prime umbers are,, 5, 7 Hece, the first four terms of the sequece are,, 5, 7 Example Fid the 0th term of the sequece defied by a = ( ) + Solutio Give a = ( ), puttig = 0, we get + 0 ( 0 ) a 0 = = = 0 + Example Fid the th ad th terms of the sequece defied by a =, whe is eve +, whe is odd Solutio As is odd = + = + = 69 + = 70; ad as is eve, a = = = 96 Example 5 Fid the first five terms of the sequece give by a =, a = + a ad a = a +5 for > Also write the correspodig series Solutio Here a =, a = + a = + = 5 Give a = a + 5 for >, puttig =,, 5, we get a = a + 5 = = 5 a = a + 5 = = 5 a 5 = a + 5 = = 75 Hece, the first five terms of the give sequece are, 5, 5, 5, 75 The correspodig series is
4 00 MATHEMATICS XI Example 6 The Fiboacci sequece is defied by a = a =, a = a + a for >, fid a a for =,,,, 5 Solutio Give a = a = ad a = a + a for > Puttig =,, 5 ad 6 i (i), we get a = a + a = + =, a = a + a = + =, a 5 = a = + = 5 ad a 6 = a 5 + a = 5 + = 8 Puttig =,,, ad 5 i a +, we get a ie a a a a5 a6,,,, a a a a a5, 5, 8,, 5 ie,, 5, 8, 5 + Example 7 (i) Fid the first terms of the series Σ ( ) + (ii) The sum of terms of a series is + for all values of Fid the first terms of the series Also fid its 0th term Solutio (i) th term of the give series a = ( ) + First term = a = ( ) + = (i) Secod term = a = ( ) + = 9 Third term = a = ( ) + = 7 Hece, first terms of the give series are,, 9 7 (ii) Give S = + S = ( ) + ( ) = + a = S S = + ( + ) = + Puttig =,, ad 0, we get a = + =, a = + = 6 = + = 8 ad a 0 = 0 + = Hece, the first three terms are, 6, 8 ad the 0th term is Example 8 If for a sequece, S = ( ), fid its first four terms Solutio Give S = ( ) S = ( ) a = S S = ( ) ( ) = ( ) = ( ) = Puttig =,,,, we get a = 0 =, a = = = = 6 ad a = = 08 Hece, the first four terms of the sequece are,, 6, 08 Example 9 (i) Write (k + ) i expaded form k= (ii) Write the series i sigma otatio +
5 SEQUENCES AND SERIES 0 Solutio (i) Puttig k =,,,,, i (k + ), we get, 5, 0, 7,, + Hece, (k + ) = ( + ) k= (ii) We see that kth term of series = k k + Hece, the give series ca be writte as k = + k + EXERCISE 9 Very short aswer type questios ( to 7) : Give a example of a sequece which is ot a progressio Which term of the sequece give by a = + +, N, is 6? Write the first three terms of the sequece whose th term is give by a = ( ) 5 If a sequece is give by a =, a = + a ad a = a for > The write the correspodig series upto terms 5 Write the ext term of each of the followig sequeces : (i),,,, (ii) 5, 5, 5, 5 8, 6 Write the ext term of each of the followig sequeces : (i) 0,, 6,, 0, (ii) 6, 9, 6, 7,, (iii), 5,, 0, 55, Hit (iii) 5 = +, = 5 +, 0 = +, 7 Write the eleveth term of the followig sequece :,,,, 5, 8,,,, 8 Write first 5 terms of the followig sequeces whose th terms are give by : (i) a = + 5 (ii) a = 6 (iii) a = ( ) (iv) a = ( + ) (v) a = (vi) a = + (vii) a = ( ) 5 + (viii) a = ( + 5 ) (ix) a = + 9 Fid the idicated term(s) i each of the followig sequeces whose th terms are : (i) a = ; a 7, a (ii) a = ; a 5, a 7 (iii) a = ( ) ; a 9 (iv) a = ( ) ( ) ( + ); a, a, a 0 0 Fid the 8th ad 5th terms of the sequece defied by ( + ), if is eve atural umber T =, if is odd atural umber + Fid the first five terms of each of the followig sequeces ad obtai the correspodig series : (i) a =, a = a +, (ii) a =, a = a +, for all > (iii) a =, a = a for (iv) a = a =, a = a for > If the sum of terms of a sequece is give by S = + for all N, fid the first terms Also fid its 0th term k =
6 0 MATHEMATICS XI First term of a sequece is ad the ( + )th term is obtaied by addig ( + ) to the th term for all atural umbers Fid the sixth term of the sequece Hit a + = a + ( + ) for all atural umbers 9 ARITHMETIC PROGRESSION (AP) A sequece ( fiite or ifiite) is called a arithmetic progressio (abbreviated AP) iff the differece of ay term from its precedig term is costat This costat is usually deoted by d ad is called commo differece Thus a, a,, a or a, a, is a AP iff a k + a k = d, a costat (idepedet of k) for k =,,, or k =,,, as the case may be It follows that, i a AP, a + = a + d ie ay term (except the first) is obtaied by addig the fixed umber d to its precedig term If the sequece a, a,, a is a AP, the the series a + a + + a is called a arithmetic series Geeral term of a AP Let a be the first term ad d be the commo differece of a AP, the the AP is a, a + d, a + d, ad its th term = a + ( ) d Hece, geeral term a = a + ( ) d Last term of a AP If the last term of a AP cosistig of terms is deoted by l, the l = a + ( ) d The th term from the ed of a fiite AP (i) If a, a + d, a + d, is a fiite AP cosistig of m terms, the the th term from the ed = (m + )th term from begiig = a + ( m + ) d = a + (m ) d (ii) If a, a + d, a + d, is a fiite AP with last term l, the the th term from ed = l + ( ) ( d) = l ( ) d For, whe we look at the terms of the give AP from the last ad move towards begiig we fid that the sequece is a AP with commo differece d ad first term as l, therefore, th term from the ed of the give AP = l +( ) ( d) = l ( ) d Sum of terms of a AP Let a be the first term, d the commo differece ad l the last term of a AP If S deotes the sum of its first terms, the S = (a + ( ) d) or S = (a + l) REMARKS If the sum of terms of a AP is deoted by S, the its commo differece = S S Three umbers a, b ad c are i AP iff b a = c b ie iff b = a + c Some problems ivolve, or 5 umbers i AP If the sum of the umbers is give, the i a AP, (i) three umbers are take as a d, a, a + d (ii) four umbers are take as a d, a d, a + d, a + d (iii) five umbers are take as a d, a d, a, a + d, a + d This cosiderably simplifies the calculatios of a ad d The sum of the terms equidistat from the begiig ad the ed of a AP is always the same ad equals to the sum of the first ad the last terms
7 SEQUENCES AND SERIES 0 9 If the terms of a arithmetic progressio (AP) are icreased, decreased, multiplied or divided by the same ozero costat, they remai i arithmetic progressio Proof Cosider the AP a, a + d, a + d, a + d, () (i) If each term of () is icreased by a costat k, we obtai the sequece a + k, a + d + k, a + d + k, a + d + k, ie a + k, a + k + d, a + k + d, a + k +d, which is clearly a AP whose first term is a + k ad commo differece is d (ii) If each term of () is decreased by a costat k, we obtai the sequece a k, a + d k, a + d k, ie a k, a k + d, a k + d, a k + d, which is clearly a AP whose first term is a k ad commo differece is d (iii) If each term of () is multiplied by a costat k, we obtai the sequece ak, (a + d) k, (a + d) k, ie ak, ak + dk, ak + dk, ak + dk which is clearly a AP with first term ak ad commo differece dk (iv) Whe each term of () is divided by k ( 0), we obtai the sequece a a d a d a d, +, +, +, k k k k k k k which is clearly a AP with first term a k ad commo differece d k ILLUSTRATIVE EXAMPLES Example The fourth term of AP is equal to times its first term ad seveth term exceeds twice the third term by Fid the first term ad the commo differece Solutio Let a be the first term ad d be the commo differece Now a = a a + d = a d = a (i) a 7 = a + a + 6 d = (a + d) + d = a + (ii) Solvig (i) ad (ii) simultaeously, we get a =, d = Hece the first term of the give sequece is ad commo differece is Example I a sequece, the th term is a = + Show that it is ot a AP Solutio Give a = + a + = ( + ) + a + a = ( ( + ) + ) ( + ) = ( ( + + ) + ) ( + ) Hece the give sequece is ot a AP = +, which depeds upo ad is ot costat Example I a sequece, the sum of first terms is S = P + ( ) Q where P, Q are costats Show that the sequece is a AP Fid its first term, commo differece ad 00th term Solutio Give S = P + ( ) Q, S = ( )P + ( ) ( ) Q a = S S = ( ( ))P + [ ( ) ( ) ( )]Q = P + ( ) ( ( ))Q = P + ( ) Q = P + ( )Q a + = P + Q a + a = (P + Q) (P + ( ) Q) = Q, which is costat Hece the give sequece is a AP with commo differece Q a = P ad a 00 = P + 99Q
8 0 MATHEMATICS XI Example Which term of the sequece 5,,,, is the first egative term? Solutio The give sequece is a AP with commo differece d = a = 5 Let th term of the give AP be the first egative term, the a < ( ) < 0 0 < 0 0 < 0 0 < > 0 > 0 ie > ad first term Sice 5 is the least atural umber satisfyig > = 5 Hece, 5th term of the give sequece is the first egative term Example 5 Which term of the sequece + 8 i, i, i, is (i) real (ii) purely imagiary? Solutio The give sequece is a AP with commo differece d = i ad first term a = + 8 i a (geeral term) = a + ( ) d = ( + 8 i) + ( ) ( i) = ( + ) + (8 + ) i = ( ) + (9 ) i (i) Let th term of the give sequece be real ( ) + (9 ) i is real 9 = 0 = 9 Hece, 9th term of the give sequece is real (ii) Let th term of the give sequece be purely imagiary ( ) + (9 ) i is purely imagiary = 0 = 7 Hece, 7th term of the give sequece is purely imagiary Example 6 How may terms are idetical i the two Arithmetic progressios,, 6, 8, upto 00 terms ad, 6, 9,, upto 80 terms Solutio 00th term of the AP,, 6, 8, = + (00 ) = 00 ad 80th term of the AP, 6, 9,, = + (80 ) = 0 Let terms be idetical i the two give Arithmetic progressios The sequece of idetical terms is 6,, 8, which is a AP with first term 6 ad commo differece 6 Its th term = 6 + ( ) 6 = 6 Sice the last term ie th term of the sequece of idetical terms ca t be greater tha 00, Hece, terms are idetical = NOTE If fially we get m, the = m if m is a iteger ad = a iteger just less tha m if m is ot a iteger
9 8 MATHEMATICS XI EXERCISE 9 ANSWERS,, 5, 7,,, 7, 8th 5, 5, (i) 5 (ii) 6 6 (i) 0 (ii) 6 (iii) (i) 7, 9,,, 5 (ii) 6, 5 6, 7, 6, 6 (iii) 0,, 6,, 0 (iv), 8, 5,, 5 (v),, 8, 6, (vi),, 5, 5, 6 (vii) 5, 5, 65, 5, 565 (viii) 9 75,,,, (ix), 5, 0 7 6,, (i) 65, 9 (ii) 5 9, (iii) 79 (iv) 0, 0, , (i),, 5, 7, 9; (ii),, 5, 07, ; (iii),, 6,, 0 ; ( ) (iv),,, 0, ; ( ) + 5, 9,, 7; 8 EXERCISE 9 5 (i) (ii) th (i) 0th (ii) 8th th st (i) 6r (ii) q b( r q) c( r p) 5 (i) m + p (ii) 0 6 p q (5 + 7) 8; 5 6 or 6 x = (i),, 8 (ii) ; 6 5 (i) 7 6 (ii) 5 56 (i) 5, 7, 9 (ii),, (iii) 7, 8, 9 (iv) 6, 0, (v), 6, 0, 57 (i) (i) 8080 (ii) ; Savig is importat for idividuals as it provides security agaist uexpected expeses It helps i the progress of the coutry ad the developmet of ecoomy 6 (i) 550 (ii) 75 (iii) 775; Keepig save eviromet ad coservatio of exhaustible resources i mid, the maually operated machie should be promoted so that eergy could be saved The maufacturer uses his/her wisdom to create more employmet for villagers by usig had operated machies
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