Question

If a, b, c and d are in G.P. show that .

Answer 1

Given that a,b,c and d are in G.P.

We have to prove that ( a 2 + b 2 + c 2 )( b 2 + c 2 + d 2 )= ( ab+bc+cd ) 2 (1)

Now, common ratio of G.P is,

b a = d c bc=ad (2)

The geometric mean of a,b, and c is,

b 2 =ac(3)

And the geometric mean of b,c and d is,

c 2 =bd(4)

Now, taking the R.H.S. part of equation (1),

( ab+bc+cd ) 2 = ( ab+ad+cd ) 2 [ from eqution ( 2 ) ] = ( ab+d( a+c ) ) 2 = a 2 b 2 + d 2 ( a+c ) 2 +2abd( a+c ) = a 2 b 2 + d 2 ( a 2 + c 2 +2ac )+2abd( a+c )

Solve further.

( ab+bc+cd ) 2 = a 2 b 2 + d 2 a 2 + d 2 c 2 +2ac d 2 +2 a 2 bd+2abcd = a 2 b 2 + d 2 a 2 + d 2 c 2 +2 b 2 d 2 +2 a 2 c 2 +2 b 2 c 2 = a 2 b 2 + d 2 a 2 + d 2 c 2 + b 2 d 2 + b 2 d 2 + a 2 c 2 + a 2 c 2 + b 2 c 2 + b 2 c 2 = a 2 b 2 + d 2 a 2 + d 2 c 2 + c 2 c 2 + b 2 d 2 + b 2 b 2 + a 2 c 2 + b 2 c 2 + b 2 c 2

Further solve.

( ab+bc+cd ) 2 = a 2 b 2 + d 2 a 2 + d 2 c 2 + c 2 c 2 + b 2 d 2 + b 2 b 2 + a 2 c 2 + b 2 c 2 + b 2 c 2 = a 2 ( b 2 + c 2 + d 2 )+ b 2 ( b 2 + c 2 + d 2 )+ c 2 ( b 2 + c 2 + d 2 ) =( a 2 + b 2 + c 2 )( b 2 + c 2 + d 2 ) =L.H.S

Hence, it is proved.