Problem: If the arithmetic mean of a a b b a > b > 0 a > b > 0 a b b a ​
Answer Choices:
A. 5 5
B. 8 8
C. 11 1 1
D. 14 1 4
E. none of theseSolution:
a + b 2 = 2 a b , a + b = 4 a b , a 2 + 2 a b + b 2 = 16 a b 2 a + b ​ = 2 a b ​ , a + b = 4 a b ​ , a 2 + 2 a b + b 2 = 1 6 a b
∴ a 2 − 14 a b + b 2 = 0 , a = 14 b + b 192 2 ∴ a b = 14 + 192 2 ≈ 14 + 14 2 = 14 ∴ a 2 − 1 4 a b + b 2 = 0 , a = 2 1 4 b + b 1 9 2 ​ ​ ∴ b a ​ = 2 1 4 + 1 9 2 ​ ​ ≈ 2 1 4 + 1 4 ​ = 1 4
Note. The value a b = 14 − 192 2 ≈ 14 − 14 2 = 0 b a ​ = 2 1 4 − 1 9 2 ​ ​ ≈ 2 1 4 − 1 4 ​ = 0