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Question

If a,b,c are in continued proportion, prove that
(a+b+c)(ab+c)=a2+b2+c2

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Solution

To Prove : (a+b+c)(ab+c)=a2+b2+c2

Proof : a,b,c are in continued proportion.

ab=bc=k (let)
b=ck
a=bk=(ck)k =ck2

L.H.S. =(ck2+ck+c)(ck2ck+c)

=c2(k2+k+1)(k2k+1)

=c2[(k2+1)2(k)2]

=c2[k4+2k2+1k2]

=c2[k4+k2+1]

R.H.S. =c2k4+c2k2+c2

=c2[k4+k2+1]

L.H.S = R.H.S.

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