Question

If A, B, C are in continue proportion the prove abc(a+B+C) ^3 = (ab+bc+cd)^3

Answer 1

a,b,c,d are in continued proportion

⇒  

b

a

​  

=  

c

b

​  

=  

d

c

​  

=k

⇒a=bk,b=ck,c=dk,

To prove a  

2

+b  

2

+c  

2

)(b  

2

+c  

2

+d  

2

)=(ab+bc+cd+)  

2

 

substituting a,b,c

⇒((bk)  

2

+(ck)  

2

+(dk)  

2

)(b  

2

+c  

2

+d  

2

)=(b  

2

k+c  

2

k+d  

2

k)  

2

 

⇒k  

2

(b  

2

+c  

2

+d  

2

)(b  

2

+c  

2

+d  

2

)=k  

2

(b  

2

+c  

2

+d  

2

)  

2

 

⇒(b  

2

+c  

2

+d  

2

)  

2

=(b  

2

+c  

2

+d  

2

)  

2