Formula Book Chapter 16

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  BRIGHT ENGINEERING CLASSES Degree and Diploma: Engineering*Management*Science*Commerce 9322420787*9321022089*9920155981*9702057971 16.1 Arithmetic Progressions (AP):   Arithmetic progression is is sequence of terms which increase or decrease by a constant number called common difference(d). It is a series of number in which each term after the first is obtained by adding to the preceding term a constant number. It can also be obtained by adding the difference of first two terms to the second term to get the third & so on. Thus a series in Arithmetic Progressions becomes an additive series in which the common difference can be found by subtracting each term from its preced ing one. It is denoted by ‘d’ Example: A.P = a, (a + d), (a + 2 d), (a+ 3d)……………   = 2,4,6,8,10………  = 10 -8 -6 -4 - 2 0…….  In arithmetic series terms are denoted by t 1  , t 2  ,t 3   First term is denoted by ‘a’ & the   last term by ‘d’ denotes the difference between two terms; t 2  –  t1. Finding n  th  term of A.P: The standard form of A.P as given above is a, (a+d), (a+2d) …… where a is the first term of the series & d is the common difference. They can be expressed in the following way: First term = a or a+(1-1)d. Second term = a +d or a + (2-1) d Third term = a+2d or a+(3-1) d n th  term = a + (n-1)d n th  term of A.P can be determined by using the following formula: T n  = a +(n-1)d  N = Number of terms & d is common difference Sum of the Series: Arithmetic Progression is a series of numbers arranged in an order.By using the following formulae, its sum can be determined. ]La[ 2nS(or) ]d)1n(a2[ 2nS nn    S n  denotes the sum of A.P n = number of terms a = First term, L = Last term, d = common difference= t 2  - t 1   Geometric Progression (GP): Geometric progression is a sequence whose terms increase or decrease by a constant ratio called the common ratio. Thus it is a multiplicative series, whose common ratio can be found  by dividing any term by its preceding term. Definitions: “ A   series or set of number is said to be in Geometric Progression when the ratio of any number of the set to its preceding number is always the same.”   “ A geometric progression is a set of quantities arranged in such a way that the ratio between any two consecutive terms is the same.”   This ratio is called the common ratio (C.R) & is denoted by ‘r’.   Sum of first n terms S um to n terms of a Geometric Progression whose first term is ‘a’ & common ratio ‘r’ Sum to n terms is denoted by S n. )1.......(r ar ar aS 1n321 n     The n th  term (T n )=ar  n-1  Chapter   Chapter 16 Arithmetic, Geometric and Harmonic Progressions  BRIGHT ENGINEERING CLASSES Degree and Diploma: Engineering*Management*Science*Commerce 9322420787*9321022089*9920155981*9702057971 16.2 The sum of the first n terms:  If -1<r<1 Then the sum of the infinite GP :  r aS  n  1    1r  if na 1r if 1r )1r (a  1r if r 1)r 1(a S nnn  Harmonic Progression (HP):   Harmonic Progression is nothing but the reciprocal of Arithmetic Progression. We take the reciprocal of the terms & follow the rules of A.P. First term is denoted by ‘a’. Fixed value added is called common difference and is denoted by ‘d’. Relationship in A.M., G.M. & H.M.: Let a & b be any two positive numbers: Then AM = 2 ba    GM= ab  HM=  baab2   (AM)(HM)= 2 )( ba2ab .2 GM abba              (AM). (HM) = (GM) 2  Therefore if a & b are any two positive numbers; then their A.M., G.M., & H.M. are in G.P.
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