IIT JEE , AIEEE Forum - Mathematics , Algebra

shobhita

if a,b,c,d are in HP ,then ab+bc+cd=3ad.

true or false .and how??????

ans>>>>true

sLeviN063

if abcd r in hp
=>1/a,1/b,1/c,1/d r in ap
=>1/d-1/c=1/c-1/b=1/b-1/a=(1/d-1/a)/3=k
=>c-d/cd=b-c/bc=a-b/ab=a-d/3ad=k
=>cd=c-d/k--(1) bc=b-c/bc---(2) ab=a-b/k---(3) 3ad=a-d/k--(4)
addin --(1),2,3
cd+bc+ab=(c-d+b-c+a-b)/k=a-d/k
but a-d/k=3ad frm --(4)
=>cd+bc+ab=3ad

Avinash_Bhat

NOW LET ME DRAW YOUR ATTENTION TOWARDS THE CONCLUSION :

If a₁, a₂, a₃, a₄, .............., a_n are in H.P. then :

a₁.a₂ + a₂.a₃ + a₃.a₄ + .............. + a_n-1.a_n = (n - 1).a₁.a_n

ruhi

okk here's a simple solution
ans is true.........
see a,b,c,d are in hp
dat means............
2/b = 1/a + 1/c .............(1)
2/c = 1/b + 1/d ..............(2)
simply multiply (1) nd (2).........nd solve
u ll get the relation...............
ab+bc+cd = ad
hope u got it :)

harshdeep89

an easy way

1/a,1/b,1/c,1/d in ap have common difference m

1/b -1/a =m

then,

a-b

----- = m

ab

thus ab=(a-b)/m

similarly for bc and cd

now lhs becomes

( a-b + b-c + c-d)/m =( a-d )/m

1/d -1/a =3m

therefore (a-d)/m =3ad

THE ANS IS TRUE..

harshdeep89

And Avinash's conclusion can be proved in similar manner

Ask & Discuss Questions with Community & Experts