Let a and b be the two numbers.
Let A and G be the A. M. and G. M. respectively between a and b
∴ A > G
Example 1. The arithemetic mean between two numbers is 34 and their geometric meanis 16. Find the numbers.
Solution : Let the numbers be a and b.
Since A. M. between a and b is 34,
.....(1)
Since G. M. between a and b is 16,
∴ √ab = 16 or, ab = 256
we know that (a – b)^{2} = (a + b)^{2} – 4 ab ......(2)
= (68)^{2} – 4 × 256
= 4624 – 1024 = 3600
∴ a – b = √3600 = 60 ....... (3)
Adding (1) and (3), we get, 2a = 128
∴ a = 64
Subtracting (3) from (1), are get
2b = 8 or, b = 4
∴ Required numbers are 64 and 4.
Example 2. The arithmetic mean between two quantities b and c is a and the twogeometric means between them are g_{1} and g_{2}. Prove that
Solutions: The A. M. between b and c is a
Again g_{1} and g_{2} are two G. M.'s between b and c
∴ b, g_{1}, g_{2}, c are in G. P.
If r be the common ratio, then
c = br^{3}
or
= bc (2a) [since b + c = 2a]
= 2abc
Example 3. If one geometric mean G and two arithmetic means p and q be inserted between two quantities, show that : G^{2} = (2p – q) (2q – p)
Solutions: Let the two quantities be a and b, then
G = √ab or, G^{2} = ab ...... (1)
Also p and q are two A. M.'s betweena and b
∴ a, p, q, b are in A. P.
∴ p – a = q – p and q – p = b – q
∴ a = 2p – q and b = 2q – p
∴ G^{2} = ab = (2p – q) (2q – p)
Example 4. The product of first three terms of a G. P. is 1000. If we add 6 to its secondterm and 7 to its 3rd term, the three terms form an A. P. Find the terms of the G. P.
Solutions: Let t_{1} = a/r, t_{2} = a and t_{3} = ar be the first three terms of G. P.
Then, their product = a/r. a.ar = 1000 or a^{3} = 1000 or a = 10
By the question, t_{1}, t_{2} + 6, t_{3} + 7 are in A. P. ...... (1)
i.e, a/r, a + 6, ar + 7 are in A.P.
∴ (a + 6) – a/r = (ar + 7) – (a + 6)
or (a + 6) = a/r + (ar + 7)
or, 2(10+6) = 10/r + (10r + 7)
or, 32r = 10 + 10 r^{2} + 7r
or, 10r^{2} – 25r + 10 = 0
= 2, 1/2
When a = 10, r = 2. then the terms are 10/2, 10(2) i.e. 5,10,20
When a = 10, r = 1/2, then the terms are 10(2), 10, 10(1/2)
i.e., 20, 10, 5
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