Let a and b be the numbers such that A, G are A.M. and G.M. respectively between them.
A.M. of a and b =

a+b

2

i.e., A = a+

b

2

⇒ a + b = 2A ... (i)
Also, G.M. of a and b = √ab i.e., G = √ab
⇒ G2 = ab .....(ii)
We know that, (a–b)2 = (a+b)2 – 4ab
⇒ (a–b)2 = (2A)2–4G2 [By using (i) and (ii)]
⇒ (a–b)2 = 4A2–4G2
⇒ a - b = 2 √A2−G2 ... (iii)
Adding (i) & (iii), we get
2a = 2A + 2√A2−G2 ⇒ a = A + √(A−G)(A+G)
Subtracting (iii) from (i), we get
2b = 2A - 2 √A2−G2 ⇒ b = A - √(A−G)(A+G)
Hence, the numbers are A ± √(A−G)(A+G)