Progressions. ILLUSTRATION 1 11, 7, 3, -1, i s an A.P. whose first term is 11 and the common difference 7-11=-4.

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1 Progressios SEQUENCE A sequece is a fuctio whose domai is the set N of atural umbers. REAL SEQUENCE A Sequece whose rage is a subset of R is called a real sequece. I other words, a real sequece is a fuctio with domai N ad the rage a subset of the set R of real umbers. PROGRESSIONS: It is ot ecessary that the terms of a sequece always follow a Certai patter or they are described by some explicit formula for the th term. Those sequeces whose terms follow certai patters are called progressios. ILLUSTRATION, 7, 3, -, i s a A.P. whose first term is ad the commo differece ILLUSTRATION Sow that the sequece <a > defied by a + is ot a A.P. PROPERTIES OF AN ARITMETIC PROPERTY I: If a is the first term ad d the commo differece of a A.P., the its th terms a is give by a a+(-)d PROPERTY II: A sequece is a A.P iff its th term is of the form A+B i.e. a liear expressio i. The commo differece i such a case is A i.e. the coefficiet of. PROPERTY III: If a costat is added to or subtracted from each term of a A.P., the the resultig sequece is also A.P. with the same commo differece. PROPERTY IV: If each term of a give A.P. is multiplied or divided by a o-zero costat k, the the resultig sequece is also a A.P. with commo differece kd or d/k, where d is the commo differece of the give A.P. PROPERTY V: I a fiite A.P. the sum of the terms equidistat from the begiig ad ed is always same ad is equal to the sum of first ad last term i.e. a k + a -(k-) a + a for all k,, 3,, -. PROPERTY VI: Three umbers a, b, c are i A.P. iff ba+c. PROPERTY VII: If the terms of a A.P. are chose at regulr itervals the they from a A.P. PROPERTY VIII: If a, a + ad a + are three cosecutive terms of a A.P., the a + a +a +. INSERTION OF ARITMETIC MEANS If betwee two give quatities a ad b we have to isert quatities A, A,,A such that a, A, A,..A, b form a A.P., the we say that A, A,.,A are arithmetic meas betwee a ad b. INSERTION OF ARITMETIC MEANS BETWEEN a AND b. Let A, A,.,A be arithmetic meas betwee two quatities a ad b. The, a, A, A,.,, Ab is a A.P. Let d be the commo differece of this A.P. Clearly, it cotais (+) terms. b (+)th term

2 b a+(+) d b a d + Now, A a + d b a A A a + + ( b a) A a + d A a + + ( b a) A a + d A A a + + () Betwee ad 3 are iserted m arithmetic meas so that the ratio of the 7 th ad (m-)th meas is 5 : 9. Fid the value of m. GEOMETRIC PROGRESSION A sequece of o-zero umbers is called a geometric progressio (abbreviate as G.P.) if the ratio of a term ad the term precedig to it is always a costat quatity. The costat ratio is called the commo ratio of the G.P. I other words a sequece, a, a, a 3,.,a. is called a geometric progressio if a + costat for all N. a PROPERTIES OF GEOMETIC PROGRESSIONS I this sectio, we shall discuss some properties of geometric progressios ad geometric series PROPERTY I: If all the terms of a G.P. be multiplied or divided by the same o-zero costat, the it remais G.P. with the same commo ratio. PROPERTY II: The reciprocals of the terms of a give G.P. form a G.P. PROPERTY III: If each terms of a G.P. be raised to the same power, the resultig sequece also forms a G.P. PROPERTY IV: I a fiite G.P the product of the terms equidistat from the begiig ad the ed is always same ad is equal to the product of the first ad the last term. PROPERTY V Three o-zero umbers a, b, c are i G.P. iff b ac PROPERTY VI If the terms of a give G.P. are chose at regular itervals, the the ew sequece so formed also forms a G.P. PROPERTY VII If a, a, a 3., a, be a G.P. of ozero o egative terms, the log a, loga,, loga,. is a A.P ad vice versa. () Fid all the complex umbers x ad y such that x, x+y, x+y are i A.P. ad (y+), xy + 5, (x+) are i G.P. Also, fid the progressio.

3 m () If mth term of a G.P.is m ad th term is, the prove that its rth term is SUM OF TERMS OF A G.P. TEOREM To prove that the sum of terms of a G.P. With first term a ad commo ratio r is r r give by S a or, S a, r. r r () Let S deote the sum of the terms of a ifiite G.P. ad σ deote the sum of the squares of the S σ terms. Show that the sum of the first terms of this geometric progressio is give by S S + σ () Fid the geometric progressio of real umber such that the sum of its first four terms is equal to 30 ad the sum of the squares of the first four terms is 340. (3) Prove that... is ot a prime umber. 9 times r r m / m INSERTION OF GEOMETRIC MEANS BETWEEN TWO GIVEN NUMBERS A AND b. Let G, G, G be geometric meas betwee two give umbers a ad b. The, A, G, G,., G, b is a G.P. cosistig of (+) terms. Let r be the commo ratio of this G.P. The, B (+)th term ar + r + b a r(b/a) /+ /( + ) b G ar a, a /( ) b..., + G ar a a ( + ) b G ar a a AN IMPORTANT PROPERTY OF GEOMETRIC GEANS TEOREM: If geometric meas are iserted betwee two quatities, the the product of geometric meas is the th power of the sigle geometric mea betwee the two quatities. SOME IMPORTANT PROPERTIES OF ARITMETIC AND GEOMETRIC MEANS BETWEEN TWO GIVEN QUANTITIES. PROPERTY I If A ad G are respectively arithmetic ad geometric meas betwee two positive umber a ad b, the A > G. PROPERTY II If A ad G are respectively arithmetic ad geometric meas betwee two positive quatities a ad b, the the quadratic equatio havig a, b as its roots is x Ax + G 0 PROPERTY III If A ad G be the A.M. ad G.M. betwee two positive umbers, the the umbers are A ± A G

4 () If oe geometric mea G ad two arithmetic meas A ad A be iserted betwee two give quatities, prove that G (A A ) (A -A ). () m The A.M. betwee m ad ad the G.M. betwee a ad b are each equal to. Fid m ad m + i terms of a ad b. ARITMETICO-GEOMETRIC SERIES Let a, (a + d)r, (a+d)r, (a+3d)r 3, be a arithmetico geometric sequece. The, a+(a+d)r+(a+d)r +(a+3d)r 3 + is a arithmetico geometric series. 3 4 ILLUSTRATION Fid the th term of the series SUM OF TERMS OF AN ARITMETICO-GEOMETRIC SEQUENCE TEOREM The sum of terms of a arthmetico-geometric sequece a, (a + d) r, (a+d)r, (a+3d)r 3, is give by a ( r ) { a + ( ) d} r + dr, whe r S r r r [ a + ( ) d], whe r + + EXAMPLE Show that the sum of the series to umber accordig as is eve or odd. terms is a eve or a odd ARMONIC PROGRESSION DEFINITION A sequece a, a,.,, a. Of o-zero umber is called a armoic progressio, if the sequece,,,...,,... is a Arithmetic progressio. a a a3 a th TERM OF A P The th term of a.p is the reciprocal of the th term of the correspodig A.P. Thus, if a, a, a 3,,, a is a P ad the commo differece of the correspodig AP is d i.e. d, the a a+ a + ( ) d a If a, b, c are i P, the,, ac are i AP. Therefore + b a b c b a c a c () If S, S ad S 3 deote the sum up to (>) terms of three o-costat sequece i A.P., whose first SS3 SS SS3 terms are uity ad commo differeces are i.p., prove that S S + S3 () 4 If a, b, c are i P ad a > c, show that + > b c a b a c (3) Let a, b, c be positive real umbers. If a, A, A, b are i A.P., a, G, G, b are i G.P. ad a,,, b are i.p., show that GG A + A ( )( a + b) + 9ab PROPERTIES OF ARITMETIC, GEOMETRIC AND ARMONIC MEANS BETWEEN TWO GIVEN NUMBERS Let A, G ad be arithmetic, geometric ad harmoic meas of two positive umbers a ad b. The. ab A, G ab ad. These three meas possess the followig properties :

5 ab PROPERTY A G. PROPERTY A, G, form a GP i.e, G A. PROPERTY3 The equatio havig a ad b as its roots x Ax + G 0 PROPERTY4 If A, G, are arithmetic, geometric ad harmoic meas betwee three give umbers a, b ad c, the the equatio havig a, b, c as its roots is 3 3 3G 3 x 3Ax + x G 0 () + + For what value of, is the harmoic mea of a ad b? () If A, A ; G, G ;, be two A.M. s, G.M s ad.m.s betwee two umber a ad b, the prove that : (3) If,,., be harmoic meas betwee a ad b ad is a root of the equatio x (-ab)- x(a +b )-(+ab)0, the prove that ab(a-b) + r r + METOD OF DIFFERENCES: Sometimes the th term of a sequece or a series caot be determied by the methods discussed i the earlier sectios. I such cases, we use the followig steps to fid the th term T of the give sequece. STEP Obtai the terms of the sequece ad compute the differeces betwee the successive terms of the give sequece. If these differeces are i A.P, the take T a + bc +c, where a, b, c are costats. Determie a, b, c by puttig,, 3 ad puttig the values of T, T, T 3. STEP If the successive differeces computed i step are i G.P. with commo ratio r, the take T ar - + b+c. STEP 3 If the differeces of the differeces computed i step are i A.P., the take T a 3 + b + c + d ad fid the values of costats a, b, c, d. STEP 4 If the differeces of the differeces computed i step are i G.P. with commo ratio r, the take T ar - + b + c + d The followig examples will illustrate the above procedure. () Sum the followig series to terms : () If... 3 ad, + 3 ' ( ) ( )( ) ( )( 3.3 show that (3) 4 3 Fid the sum of first terms of the series whose th term is ( ) ( ) (4) ( + )( + )( + 3) If Tr, where T r deotes the rth term of the series. Fid, lim. r r r (5) Sum the series to terms: ( + x)( + x) ( + x)( + 3 x) ( + 3 x)( + 4 x)

SEQUENCE AND SERIES NCERT

SEQUENCE AND SERIES NCERT 9. Overview By a sequece, we mea a arragemet of umbers i a defiite order accordig to some rule. We deote the terms of a sequece by a, a,..., etc., the subscript deotes the positio of the term. I view of