Illustration 1: Let ‘a’ and ‘b’ be positive real numbers. If a, A1, A2, b are in arithmetic progression, a, G1, G2, b are in geometric progression and a, H1, H2, b are in harmonic progression, show that
G1G2/ H1H2 = (A1 + A2)/ (H1 + H2) = (2a+b)(a+2b)/9ab. (2002)
Solution: We are given that a, A1, A2, b are in A.P
Hence, A1+A2 = a+b
Now, a, G1, G2, b are in G.P
So, G1G2 = ab
Similarly, as a, H1, H2, b are in H.P
So, H1 = 3ab/ (2b+a)
And H2 = 3ab/ (b+2a)
Hence, 1/H1 + 1/H2 = 1/a + 1/b
So, (H1 + H2)/ H1.H2 = (A1+A2)/G1.G2 = 1/a + 1/b
Now, (G1G2)/ H1.H2 =
= (2a+b) (a+2b)/ 9ab
Thus combining the obtained results, we get
(G1G2)/ H1.H2 = (A1+A2)/(H1 + H2) = (2a+b) (a+2b)/ 9ab
Illustration 2: Let A1, H1 denote the arithmetic and harmonic means of two distinct positive numbers. For n ≥ 2, let An-1, Hn-1 have arithmetic and harmonic means as An and Hn respectively. Then which of the following statements is correct? (2007)
(a) 1. H1 > H2 > H3 > ….
2. H1 < H2 < H3 < ….
3. H1 > H3 > H5 > …. and H2 < H4 < H6 < ….
4. H1 < H3 < H5 < …. and H2 > H4 > H6 > ….
(b) 1. A1 > A2 > A3 > ….
2. A1 < A2 < A3 < ….
3. A1 > A3 > A5 > …. and A2 < A4 < A6 < ….
4. A1 < A3 < A5 < …. and A2 > A4 > A6 > ….
Solution: H1 is the harmonic mean between two numbers ‘a’ and ‘b’.
A1 = (a+b)/2
H1 = 2ab/ (a+b)
Hn = 2An-1H n-1 / (An-1 + H n-1)
An = (An-1 + H n-1)/ 2
A2 is A.M of A1 and H1 and A1 > H1
Hence, A1 > A2 > H1
A3 is A.M of A2 and H2 and A2 > H2
So, A2 > A3 > H2
…… …… …….
Hence, A1 > A2 > A3 > ……
Now, as we have stated above A1 > H2 > H1, A2 > H3 > H2
So, H1 < H2 < H3 < ….
Illustration 3: Prove that three quantities a, b, c are in A.P., G.P., or H.P. iff
(a-b)/(b-c) = a/a, a/b or a/c respectively.
Solution: If a, b and c are in A.P, then
b-a = c-b
This can be written as a, b and c are in A.P. iff b-a = c-b
iff (a-b)/(b-c) = 1 = a/a
Similarly, when a, b and c are in G.P,
iff b/a = c/b i.e
iff 1 - b/a = 1 - c/b
i.e. iff (a-b)/a = (b-c)/b
i.e. iff (a-b)/(b-c) = a/b
Similarly a, b, c are in H.P.
iff 1/b - 1/a = 1/c - 1/b
i.e. iff (a-b)/ab = (b-c)/bc
i.e. iff (a-b)/(b-c) = ab/bc = a/c