if-a-x-y-b-is-an-AP-and-a-p-q-b-is-a-GP-prove-that-xy-pq-with-a-b-gt-0-

Question Number 198147 by mr W last updated on 11/Oct/23

i f a, x, y, b i s a n A P a n d a, p, q, b i s a G P .

p r o v e t h a t x y ⩾ p q .

(w i t h a, b > 0)

Answered by AST last updated on 11/Oct/23

p^{2} = a q; q^{2} = p b \Rightarrow p^{2} q^{2} = a b p q \Rightarrow p q = a b

2 x = a + y; 2 y = x + b \Rightarrow 4 x y = (a + y) (x + b)

3 x y = a x + a b + y b \overset{?}{⩾} 3 p q = 3 a b

\Leftrightarrow a x + y b ⩾ 2 a b

\Leftrightarrow a (a + d) + (a + 2 d) (a + 3 d) ⩾ 2 (a) (a + 3 d)

a^{2} + a d + a^{2} + 3 a d + 2 a d + 6 d^{2} ⩾ 2 a^{2} + 6 a d

\Leftrightarrow d^{2} ⩾ 0 (t r u e) . . H e n c e 3 x y ⩾ 3 a b \Rightarrow x y ⩾ a b = p q

E q u a l i t y h o l d s w h e n d = 0 \Rightarrow a = b = x = y = p = q

Commented by mr W last updated on 11/Oct/23

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