Algebra theory weighted means examples
Ex1: Prove that ( a + b 2 ) a + b ≥ a b ⋅ b a \left( \frac{a + b}{2} \right)^{a + b} \geq a^b \cdot b^a a + b 2 > ( a b b a ) 1 a + b \frac{a + b}{2} > (a^b b^a)^{\frac{1}{a + b}} WAM ≥ WGM WAM ≥ WGM { a b , b a } \text{WAM} \geq \text{WGM} \quad \left\{ \text{a}^b, \text{b}^a \right\} a , a , a , … a, a, a, \ldots — b b times b , b , b , … b, b, b, \ldots — a a times WAM = a 1 w 1 + a 2 w 2 + … + a n w n w 1 + w 2 + … + w n = a b + b a a + b \text{WAM} = \frac{a_1 w_1 + a_2 w_2 + \ldots + a_n w_n}{w_1 + w_2 + \ldots + w_n} = \frac{a b + b a}{a + b} WGM = ( a 1 w 1 a 2 w 2 … a n w n ) 1 w 1 + w 2 + … + w n = ( a b b a ) 1 a + b \text{WGM} = (a_1^{w_1} a_2^{w_2} \ldots a_n^{w_n})^{\frac{1}{w_1 + w_2 + \ldots + w_n}} = (a^b b^a)^{\frac{1}{a + b}} WAM ≥ WGM \text{WAM} \geq \text{WGM} a b + b a a + b ≥ ( a b b a ) 1 a + b \frac{a b + b a}{a + b} \geq (a^b b^a)^{\frac{1}{a + b}} 2 a b a + b ≥ ( a b b a ) 1 a + b (1) \frac{2 a b}{a + b} \geq (a^b b^a)^{\frac{...