11TH•Mathematics•Sequences And Series•Question 14

Overview

First three of the four numbers $a,\ b,\ c,\ d$ are in A.P., and the next three are in H.P., show that $ad = bc$.

(a,b,c) in A.P. ⇒ (2b=a+c)

(b,c,d) in H.P. ⇒ (\dfrac{1}{b},\dfrac{1}{c},\dfrac{1}{d}) in A.P. ⇒ (2\cdot\dfrac{1}{c}=\dfrac{1}{b}+\dfrac{1}{d})

[ \dfrac{2}{c}=\dfrac{d+b}{bd}\implies 2bd=c(b+d) ]

Need (ad=bc). From (2b=a+c) we get (a=2b-c). Substitute and use the H.P. relation to verify (ad=bc).

Proved.

forum

menu_bookLesson...

Start discussion

keyboard_arrow_down

Discuss theoretical concepts, mathematical formulas, and architectural foundations.

No comments yet. Be the first to start the discussion!