Saturday, June 7, 2008

Ch. 6. Sequences and Series - 4 - Harmonic Progression

Harmonic progression - Revision points

A sequence a1, a2, a2,…,an of non-zero numbers is called a Harmonic progression if the sequence 1/a1,1,a2,..,1/an,.. is an A.P.

Example: The sequence 1,1/4,1/7,1/10,… is a H.P. because the sequence 1,4,7,10,… is in A.P.

d of the corresponding AP = 1/a2 -1/a1

an of H.P. 1/[a+(n-1)d] where a = 1/a1

Insertion of n harmonic means between two give numbers a and b

a,H1,H2,…,Hn,b are in H.P.
=> 1/a, 1/H1,1/H2,…,1/Hn,1/b are in A.P.

Let d be common difference of this A.P.
The last term in AP 1/b is the (n+2)th term.

So 1/b = 1/a +(n+1)d
=> d = (1/b -1/a)/(n+1) = (a-b)/ab(n+1)
=> 1/H1 = (1/a) +d
1/H2 = (1/a)+2d

1/Hn = (1/a)+nd

Harmonic mean of n numers

If a1, a2, ..., an are n non-zero numbers, then the harmonic mean H of these numbers is given by

1/H = [1/a1 +1/a2+…+1/an]/n

Properties of arithmetic, geometric and harmonic mean between two numbers a, b

A = (a+b)/2
G = √ab
H = 2ab/(a+b)

1. A≥G≥H
2. A,G,H form a GP, i.e., G² = AH
3. The equation x²-2Ax+G² has as its roots a and b.
4. The equation x³-3A x²+3G³x/H-G³ = 0 has as its roots a,b,c when A,G,and H are arithmetic mean, geometric mean, and harmonic mean of three numbers a,b,and c.

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