Sequence and Series Notes

Page # 2 SEQUENCE AND SERIES (ii) In an A.P., the sum of terms equidistant (ix) If for an A.P. sum of p terms is equal to from the beginning and end is constant and sum of q terms then sum of (p + q) terms is equal to the sum of first and last term. zero. (iii) Any term of an AP (except the first term) 6. GEOMETRICAL PROGRESSION (G.P.) is equal to the half of the sum of terms Geometric Progression is defined as a series in equidistant from the term i.e. which ratio between any two consecutive terms an b is constant throughout the series. This constant g 1 a  ank , k < n = 2 n k ratio is called as a common ratio. If ‘a’ is the (iv) If in a finite AP, the number of terms be odd, then its middle term is the AM between the first and last term and its sum is equal to the product of middle term and no. of terms. Some standard results (i) r  r 1 Note : a, b, c are in G.P. if  b2 = ac General Term of a G.P. : General term (nth term) of a G.P. is given by Sum of n terms of a G.P. b g n n1 = GP can be written as a + ar + ar2 + ..... Tn = arn–1 Sum of first n natural numbers n first term and ‘r’ is the common ratio, then a The sum of first n terms of an A.P. is given by 2 (ii) Sum of first n odd natural numbers Sn = e a 1 rn 1 r j= a  rTn 1 r when r < 1 n  b2 r  1g =  r 1 n2 or (iii) Sum of first n even natural numbers n  2r = = r 1 b gb g n n  1 2n  1 r 1 6 r3 = = r 1 LM n bn  1g OP MN 2 PQ 2 Sn = r 1 r 1  Tr = A  r n +B r r 1 2 n n +C r r 1 +D 1 r 1 (vii) If for an A.P. pth term is q, qth term is p then mth term is = p + q – m (viii) 7. GEOMETRICAL MEAN (G.M.) last term are called term of AP is 3 a  = 1 r then all the numbers lying between first and Tr = Ar3 + Br2 + Cr + D, then sum of n n i.e. | r | < 1) is If three or more than three terms are in G.P. (vi) If rth term of an A.P. n when r = 1 common ratio r (– 1 < r < 1 S (v) Sum of cubes of first n natural numbers n and Sn = nr when r >1 The sum of an infinite G.P. with first term a and n r2 = rT  a a (r n  1) = n r 1 r 1 Sum of an infinite G.P. n ( n+ 1) (iv) Sum of squares of first n natural numbers = Sn = If for an AP sum of p terms is q, sum of q terms is p, then sum of (p + q) term is – (p + q) . M ea ns between them.i.e. The G.M. between two given quantities a and b is G, so i.e. that a, G, b, are in G.P. b G =  G2 = ab  G = a G ab n GM’s between two given numbers If in between two numbers ‘a’ and ‘b’, we have to insert n GM G1, G2....Gn then a, G1, G2 , .... Gn, b will be in GP. The series consist of (n+2) terms and the last term is b and first  1 Geome tr ica l ar n+2– 1 = b term is a. Page # 3 SEQUENCE AND SERIES r =  G1 FG b IJ H aK 1 n 1 (vii) If a1, a2, a3 ....an is a G.P. of non zero , non negative terms, then log a1, log a2,...... ar2 = ar, G2 = = arn ......Gn or Gn = b/r 8. SUPPOSITION OF TERMS IN A G.P. (i) When no. of term be odd. then we take three terms as a/r, a, ar a Five terms as a , a , ar, ar2 r , r2 Here we take middle term as ‘a’ and common ratio as ‘r’. (ii) When no. of terms be even then we take 4 terms as : a r5 , a r3 , a r 3 , a ,ar, ar3 6 terms as : r a , ar, ar3,ar5 r log an is an A.P. and vice-versa. (viii) If a1, a2, a3....... and b1, b2, b3 ....... are two G.P.’s then a1 b1, a2b2, a3b3..... is also in G.P. 10.ARITHMETICO- GEOMETRICAL PROGRESSION (A.G.P.) If each term of a Progression is the product of the corresponding terms of an A.P. and a G.P., then it is called arithmetic- geometric progression (A. G.P.) e.g. a, ( a + d) r, ( a + 2d) r2 , ...... The general term (nth term) of an A.G.P. is Tn = [a + ( n-1) d] rn–1 To find the sum of n terms of an A.G.P. we Here we take a , ar as middle terms and r common ratio of the corresponding G.P. and common ratio as r2. 9. SOME PROPERTIES OF A G.P. (i) suppose its sum S, multiply both sides by the If each term of a G.P. be multiplied or than subtract as in following way and we get a G.P. whose sum can be easily obtained. divided by the same non zero quantity, then Sn = a + (a + d) r + (a + 2d) r2 + ....[a + resulting series is also a G.P. (n–1) d] rn–1 rSn = (ii) In an G.P., the product of two terms which ar + (a+d) r2 + ......+ [a+(n–1) d] rn are at equidistant from the first and the last term, is constant and is equal to product of first and last term. (iii) If each term of a G.P. be raised to the same power, then resulting series is also a G.P. After subtraction we get Sn (1– r) = a+ r.d + r2 d +...drn–1– [a + (n–1)d] rn After solving (iv) In a G.P. every term (except first) is GM of its two terms which are at equidistant from it. i.e. Tr = Tr k Tr  k k < r (v) In a finite G.P. , the number of terms be odd then its middle term is the G.M. of the first and last term. (vi) If the terms of a given G.P. are chosen at regular intervals, then the new sequence is also a G.P. Sn and a = + 1 r S = e j b1  rg r. d 1  r n 1 dr a + (1  r ) 2 1 r 2