Revision Notes on Harmonic Progression and Harmonic Mean
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Revision Notes on Harmonic Proqression and Harmonic Mean Let a, If a, b and c form an H. P. b and c are in H . Then 1/ a, 1/ b and 1/ c form an A. P. then 2/ b which can be simplified as b 2ac/ (a+c) If 'a' and 'b' are two non-zero numbers then the sequence a, H, b is a The n and b, if numbers H 1, Hn, , Hn are said to be harmonic means between a b are in H. P. i.e. if 1/ a, 1/ HI, 1/H2, a HI, are in E-12 Let d be the common difference of the A. P. , a—b/ (n+l) ab. 1/H2 1/ a + 2 (a —n)/ (n+l)ab, Then Thus I/HI If x 1, x2, these numbers is 1 are n non-zero given by 1/ H n (a—b) / (n+l) ab. numbers, then the harmonic mean +1/ Xn) 11 H' of 1 1 As the nth term of an A.P is given by an = a + (n-l)d, So the nth term of an H.P is given by 1/ If we have a set of weights WI, w2, wn associated with the set of values Xl, x2, El—I then the weighted harmonic mean is defined as Questions on Harmonic Progression are generally converting them into those of Arithmetic Progression. solved by xn, first If 'a' and 'b' are two positive real numbers then A.M x H.M The relation between the three means is defined as A.M > G.M > H.M If we need to find three numbers in a H. P. as l/a—d, 1/ a, l/a+d Four convenient numbers Five convenient numbers in H. P. in H. P. are are —3d, —2d, then they should be assumed 1/ a—d, l/a—d,
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