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01_Straight_Line 02_Circle 03_Permutation_and_combination 04_Complex_numbers 05_Application_of_Derivative 06_Binomial_theorem 07_Progression__Series.pdf 08_Definite_Integral.pdf 09_Indefinite_Integral.pdf 10_Probability.pdf

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    PROGRESSION AND SERIES INTRODUCTION An arrangement of numbers {al, Q, Q, ... , an, ... } according to some well defined rule or a set of rules is called a sequence. More precisely, we may define a sequence as a function whose domain is some subset of set of n} = X (say) to some other set of numbers Y. i.e. f: Y. natural numbers N of the type {1, 2, 3, The ordered set of images in Y given by {f (1), f (2), f (3)}, f (n)} is the sequence. Sequence containing finite number of terms is called a finite sequence and infinite sequence if it contains infinite number of terms. In case Y = R, the sequence is a real sequence and if Y = C, the sequence is a complex sequence. If [al, Q, Q, , an, ... } is a sequence, then the expression al + a2 + a3 + + an + is called the series associated with the sequence. PROGRESSION A sequence is said to be a progression if its terms numerically increase (or numerically decrease) continuously. ARITHMETIC PROGRESSION A progression {al, Q, Q, .., an, ... } is called an arithmetic progression (A. P.) if a2 - al = - a2 = = an - an-I In general ami - an = constant (say, d) n N. The constant difference d is called common difference of A.P. If the first term al of the A.P. be denoted by a then the A.P. is {a, a + d, a + 2d, ...}. Clearly, the general term of A.P. is given by an = a + (n- 1) d, n e N. General characteristics of A.P. 2. 3. 4. 5. 6. If term of any sequence in a linear expression in n, then the sequence is an A.P. If an is of the form An + B, then the common difference is A. For an A.P. {al, Q, Q, (a) {al ± k, a2 ± k, ± k, , an +k, ... } is an A.P., where kis a constant. (b) {kal, ka2, ka3, .., kan, ... } is an A.P., where kis a constant. al a2 a3 an (c) is an A.P., where 0, a constant. k (d) {ap, % q, ap+2q, ... } is an A.P. for any pand q. If {al, Q, arg, , an, ... } and {bl, b2, b3, ..., bn, ... } be two different A.P.'s then + bl, a2 + b2, an + bn, and - bl, Q - b2, an - bn, ...> are A.P. If three terms to be selected in A.P., choose a- d, a, a + d. If four terms to be selected in A.P., choose a- 3d, a- d, a + d, a + 3d. The kh term from end of an A.P. = (n + 1 - k)th term from beginning = a + (n Alternatively kh tern from end = 1+ (k- 1) (-d), where lis the last term. 7. The sum of terms equidistant from beginning and end is constant. kh term from beginning kh term from end tic = a + (k- 1) d So, tk+ TIC = 2a + (n- 1) d ( = a + l) = constant. Thus, for an A.P. {al, Q, Q, al + an = ab + an—I = + an—2 Sum of n terms of an A.P. Let Also , = an + an—I + an—2 + • • • + al Adding, we get = (al + an) + + an—I) + + an—2) + • • • + (an + al) = n{2a + (n- 1) d}
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    Then Thus, 2 n 2 Notes: Al + A 2 + A 3 + + An = n = -2k Progression and Series 1. The sum of n terms of an A.P. is a quadratic expression of the form An2 + Bn. 2. If Sn be the expression for sum of n terms, then the nth term is an = Sn- Sn_l, n > 1 and al ARITHMETIC MEAN A is said to be arithmetic mean of two numbers a and b, if a, A, b are in A.P. Thus, Note: If al, arg, n 2 an be n terms, then their statistical arithmetic mean is defined by Inserting n arithmetic means between two terms a and b Let Al, .42, A3, , An be inserted between a and b in that order such that a, Al, ,42, A3 . .. , An, bis A.P. b = (n + 2)th term = a + (n + 1) d b—a d n arithmetic means between a and b are as follows : b—a an +b 2(b —a) a(n —1) +2b A An We note that a + rd a + nd r(b — a) a + nb 2 That is, sum of n A.M. terms between a and b = n x A.M. of a and b. Example 1 : Solution: Prove that in any arithmetic progression , whose common difference is not equal to zero, the product of two terms equidistant from the extreme terms is the greater the closer these terms are to the middle term. Let { an } be the A.P., Q' be the kth term from the end = alan (k — 1) 2 c12 + (k — (an al) = alan (k — 1 ) 2 d2 + (k ak •Q' apn +d2 —1) (k —1)21 = apn +6 (k l)(n k) It is enough, if we prove Pk = ( k — l)(n — k) increasing with an increase in k from Ito — 1)d2 are 2 Pk+l . > if n GEOMETRIC PROGRESSION A progression {al, Q, Q, n —2k > 0 i.e. if k < — , an, ... } is called a geometric progression (G.P.) if
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    Progression and Series al a2 3 an (al Yi an I an+l = constant (say, r), n e N. In general an The constant ratio r is known as the common ratio of G.P. If the first term a of the G.P. be denoted by a, then the G.P. is {a, ar, ar2, ..}. Clearly the general term of A.P. is given by an = atn-l, ne N. General characteristics of G.P. , an, ... } is in G.P., then (a) {4k, 4k, a3k, , ank, ...}, O, is a G.P. al a2 a3 (b) 1 1 1 (c) al a2 a3 a k 1 an . o, is aG.P. is a G.P. (d) {ap, % q, ap+2q, ...} is a G.P. 2. 3. 4. 5. 6. 7. If {al, 22, arg, ..., an, ... } and {bl, b2, b3, al a2 a3 a a3b3,..., anbn, ... } and bl ' b2 'b3 If {al, Q, Q, viceversa. If {al, Q, arg, , bn, ... } be two different geometric progression then {albi, a2b2, are in G.P. . , an, ... } is a G.P. of positive terms then {logal, loga2, loga3, , logan, ... } is an A.P. and , an, ... } is an A.P. then for any x > 0, X If three terms to be selected in G.P., choose them as —, a, ar 3 If four terms to be selected in G.P., choose them as —, ar, ar The kh term from the end in a G.P. = (n + 1 - k)th term from beginning {zal, aan,...} is a G.P. = atn—k. Alternatively 8. 1 kh term from the end = I - , where lis the last term of the G.P. The product of terms equidistant from beginning and end is constant. kh term from beginning tic = ark kh term from end TIC = arn-k. So, tk.Tk= a2tn-l at) = constant. Thus, for a G.P. {al, Q, Q, al .an = a2.an—1 = Q. an—2 Sum of n terms of a G.P. ... a2rn—1 S n = + + arn—l Let r. Sn = + + arn—l + arn On subtracting we get a(l -r n ) 1 In fact it is advisable to use above formula in the following form 1 a(rn 1) if if
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    4 Notes: If r = 1, then the G.P. becomes a + a + a + to n terms = na Sum of infinite terms of a G.P. a(l -r n ) The sum of n terms has been obtained Sn 1 If I rl < 1, then lim rn (if tn being the nth term of the progression) (i.e., n +00 Thus sum of infinite terms of a G.P. a S=a+ ar, ar2 + 00, Irl 1, then the sum of an infinite G.P. is not defined. Gn - arn Progression and Series lim tn n 900 + 0) and then lim Sn n +00 a 1 (ii) If Snbe the expression for sum of n terms, then the nth term Sn- Sn-l, n > 1 and ti = Sl. GEOMETRIC MEAN A positive number G is said to be geometric mean of two positive numbers a and b, if a, G, b are in G.P. G A.GB. Thus, Note: If al, Q, arg, . , an be n terms then their statistical geometric mean is defined as G- (ala2Q ...an )l/n Inserting n geometric means between two terms a and b Let Gl, G2, Q, ..., Gn be inserted between a and bin order such that a, Gl, G2, 1 Then b = (n + 2)th term an +1 r a Thus, n geometric means a and b are as follows: 1 1 , Gn, bis G.P. - ar ar a a a a a a -a a a a 3...Gn 2 3 k an +1 bn+l n—l 2 an +1 bn+l 3 an +1 bn+l k a 1 an +1 bn+l We wrote that That is, product of n G.M. between a and b = r} h power of G.M. between a and b. Example 2 : If three successive terms of a G.P form the sides of a triangle then show that common ratio 'r' satisfies the inequality —($—1) < r < —(6+1).
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    Solution: Progression and Series 2-r-l < o 5 Let a, ar, ar2 be the terms. For triangle formation the necessary and sufficient condition is the sum of any two sides be larger than the third side. Hence ar + ar2 > a ( assuming r < 1) r 2 -I-r -1 ( since > O) 2 US-I 2 Consider r > 1 2 2 then 1 2 2 Hence the result. Alternatively: From (i) if ris replaced by 6-1 1 we will have
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    6 On subtraction, we get On subtraction, we get 2 2 HARMONIC PROGRESSION Progression and Series 2 00) = 4 The sequence {al, Q, Q, , an, ...} of non-zero terms is said to be a harmonic progression (H.P.), if the sequence formed by the reciprocals of its terms is an A.P. That is, the sequence 1 1 1 A.P. clearly, the standard form of H.P. is Notes: 1 1 1 al a2 a3 1 1 an where a is an 1 al 1. 2. 3. The general term of the harmonic progression {al, Q, Q, 1 1 and d a2 al ... } is given bya Corresponding to every H.P. there is an A.P. and vice versa. Therefore problems in H.P. can generally be solved with reference to the corresponding formulas of A.P. There is no formula for finding the sum of n terms of a H.P. HARMONIC MEAN A number H is said to be harmonic mean of two non-zero numbers a and b, if a, H, b are in H.P. Thus Notes: 2ab 1 1 1 1 . an be n non-zero terms then their statistical harmonic mean is defined as 1 1 1 1 n al a2 1 a Inserting n harmonic means between two terms a and b Let HI, H2, H2, , Hn be inserted between a and b in that order such that a, HI, H2, , Hnbare in H.P. Then 1 1 1 1 1 1 1 — are in A.P. a-b ab(n 1) Thus, n harmonic means between a and b are as follows : 1 Hi 1 1-12 1 1-13 a—b 1 1 a ab(n +1) a 1 1 a 1 1 a 2(a — b) ab(n + l) ab(n -k 1) ab(n + l) a + nb ab(n + 1) ab(n 1)
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    Progression and Series 1 1 Example 4 : Solution: 1 1 _ — + kd a 1 1 a k(a — b) ab(n l) n(a —b) ab(n 1) 1--1k ab(n l) ab(n + l) na+b If a, b, c be in H.P. prove that 1 1 1 1 1 1 abc b ca 2 a, b, c are in H.P. -s ac b2 1 1 1 1 Now b 1 1 1 1 1 1 2 1 b 1 7 1 C 2 1 b 1 1 a 2 2 b 2 1 c 4 ac 1 4 ac 1 a 1 3 1 4 ac Relation between A.M., G.M., H.M. If a and b are two positive numbers, then (i) A, G, and H are in G.P., i.e., & = AH (ii) A > G > H. Equality holds if and only if a = b. (iii) If al, Q, arg, . an are n positive numbers, then for their statistical means A > G > H. Equality holds if and only if al = = an. SPECIAL SEQUENCE 1. 2. 3. 4. Sum of first n natural numbers 2 Sum of squares of first n natural numbers 12 +22 +32 + + n2 6 Sum of cubes of first n natural numbers 13 +23 +33 + + n3 2 2 Sum of sequences using sigma notation or Exn. If a sequence is characterized by {xn}. Then we write Sn = + + + xn = r=l Under this notation the above three summations can be denoted by in, Erf and Erf respectively. Suppose the general term of a particular sequence {xn} is given by xn = an3 + bn + cn+ d + kp , where a, b, c, d, k, p are constants. Then sn = = + bn2 + + d = + bEn2 + + Ed + kip 2 a 2 6 2 p(pn -1) p-l + pn, which is a G.P.]
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    8 Summation of series using method of difference Progression and Series 1. 2. Consider a series sn = al + + ar3 + + an. Let an - an-I = n—l If {t , tn_l form an A.P. or a G.P. then we find Sn as following: sn = al + + + + an-I + an sn = al + ar2 + + an-2 + an—I + an Subtracting, we get O = al + — al) + — + + (an — an—I)} — an an = al + { tl + t2 + + tn_l where Etn_l is can be easily found as it is either an A.P. or a G.P. of n- 1 terms. Now, the desired sum Sn can be calculated by Sn = Dan. Consider a series al + Q + a3 + + an. We try to express the general term as the difference of two terms of some other series, i.e., an = bn_l bn for some series {bn} Hence, al + + + • • • + an = (b2 bl) + (b3 b2) + (b4 — b3) + • • • + ( bn_l bn) = bn+l Example 5 : Solution: If a, b, c are positive real numbers, then prove that 744 4 7 1 + ab + a + b + c + abc + + bc > (ab. a. b. c. abc. ac. bc)l/7 (using AM > GM) 4 4 4 1/7 4 4 4 1/7 + > 77 (a4. b4. c4).

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