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If a,b,c,d are in hp then ab+bc+cd= A 3ad B 3bc C 3(ad+bc) D 3(a+b+c+d)

If a,b,c,d are in hp then ab+bc+cd=
A 3ad
B 3bc
C 3(ad+bc)
D 3(a+b+c+d)

Grade:11

1 Answers

Arun
25750 Points
5 years ago
 
a,b,c,d are in H.P
so (1/a),(1/b),(1/c),(1/d) are in A.P
d=(1/b)-(1/a)=(1/c)-(1/b)=(1/d)-(1/c)=x..........eq(1)
now (1/b)-(1/a)=(1/c)-(1/b)=(1/d)-(1/c)
=(a-b)/ab=(b-c)/bc=(c-d)/cd=(a-b+b-c+c-d)/(ab+bc+cd) by addendo
=(a-d)/(ab+bc+cd)
taking ad common
(1/b)-(1/a)=ad{(1/d)-(1/a)}/(ab+bc+cd)......... eq(2)
as we know
an=a1+(n-1)d
put n=4
a4=a1+(4-1)d
putting a4=1/d , a1=1/a , d=x
1/d=1/a+3x
put this in eq(2)
(1/b)-(1/a)=ad{(1/a)+3x-(1/a)}/(ab+bc+cd)
x=3adx/(ab+bc+cd)
ab+bc+cd=3ad

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