Algebra> if a, b, c, d are in HP, then prove that ...

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

Ipsita Padhee , 10 Years ago

Grade upto college level

6 Answers

san

Last Activity: 10 Years ago

(2ac/(a+c))=b ...1 and (2bd/b+d))=c ...2 [property used=harmonic mean] multiplying one and two 4abcd=bc(a+c)(b+d) 4abcd=ab2c + abcd + b2c2 + c2bd 3abcd=ab2c + b2c2 + c2bd---(3) dividing (3) by bc 3ad=ab+bc +cd can u help me as well my ques is in thermal physics sec title is doubt thanx

Subham Sen

Last Activity: 10 Years ago

(1/b)-(1/a)=(1/c)-(1/b)=(1/d)-(1/c)=x(say).........................................(1) =>(a-b)/ab=(b-c)/bc=(c-d)/cd=(a-b+b-c+c-d)/(ab+bc+cd)....................[By addendo] so, (1/b)-(1/a)=(a-d)/(ab+bc+cd) =>(1/b)-(1/a)=ad{(1/d)-(1/a)}/(ab+bc+cd) [taking `ad` common] ......................................(2) now t(n)=a+(n-1)d or, 1/d = (1/a)+(4-1)x [since , a,b,c,d are in h.p.] [also from eqn (1), d=(1/b)-(1/a)=x] so, 1/d = (1/a)-3x Now substituting this value in eqn (2) we get , (1/b)-(1/a)=ad{(1/a)+3x-(1/a)}/(ab+bc+cd) or, x = 3adx/(ab+bc+cd) thus ab+bc+cd=3ad thanx n hey , the sum was cool...........

Abdul Moiz

Last Activity: 7 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

a_n=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

Jatin

Last Activity: 6 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

a_n=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

Ankit

Last Activity: 6 Years ago

a b c d are in hp then hp formula. Now using formula b=2ac/a+c. b=2ac/a+c. equation 1 . C=2bd/b+d equa...2 . Now eq(1) and eq(2) are multiple. Then. bc= 4abcd/(a+c).(b+d) bc is common so bc cut . 4ad=(a+c).(b+d). 4ad=ab+ad+bc+cd. 4ad-ad=ab+bc+cd. 3ad=ab+bc+cd......the proove it. thanku