25. If a, b, c and d are in GP. show that(Đ°Đł + b2 + c*) (b? + c2

1.	25. If a, b, c and d are in GP. show that(Đ°Đł + b2 + c*) (b? + c2 + d2)-(ab + bc + cd),
Answer» a,b,c,dare in G.P. Therefore, bc=ad… (1) b2=ac… (2) c2=bd… (3) It has to be proved that, (a²+b²+c²) (b²+c²+d²) = (ab+bc–cd)² R.H.S. = (ab+bc+cd)² = (ab+ad+cd)²[Using (1)] = [ab+d(a+c)]² =a²b²+ 2abd(a+c) +d²(a+c)² =a²b²+2a²bd+ 2acbd+d²(a²+ 2ac+c2) =a²b²+ 2a²c²+ 2b²c²+d²a²+ 2d²b²+d²c²[Using (1) and (2)] =a²b²+a²c²+a²c²+b²c²+b²c²+d²a²+d²b²+d²b²+d²c² =a²b²+a²c²+a²d²+b2×b2+b²c²+b²d²+c²b²+c²×c²+c²d² [Using (2) and (3) and rearranging terms] =a²(b²+c²+d²) +b²(b²+c²+d²) +c²(b²+c²+d²) = (a²+b²+c²) (b²+c²+d²) = L.H.S. ∴ L.H.S. = R.H.S.

25. If a, b, c and d are in GP. show that(Đ°Đł + b2 + c*) (b? + c2 + d2)-(ab + bc + cd),

Answer»

a,b,c,dare in G.P.

Therefore,

bc=ad… (1)

b2=ac… (2)

c2=bd… (3)

It has to be proved that,

(a²+b²+c²) (b²+c²+d²) = (ab+bc–cd)²

R.H.S.

= (ab+bc+cd)²

= (ab+ad+cd)²[Using (1)]

= [ab+d(a+c)]²

=a²b²+ 2abd(a+c) +d²(a+c)²

=a²b²+2a²bd+ 2acbd+d²(a²+ 2ac+c2)

=a²b²+ 2a²c²+ 2b²c²+d²a²+ 2d²b²+d²c²[Using (1) and (2)]

=a²b²+a²c²+a²c²+b²c²+b²c²+d²a²+d²b²+d²b²+d²c²

=a²b²+a²c²+a²d²+b2×b2+b²c²+b²d²+c²b²+c²×c²+c²d²

[Using (2) and (3) and rearranging terms]

=a²(b²+c²+d²) +b²(b²+c²+d²) +c²(b²+c²+d²)

= (a²+b²+c²) (b²+c²+d²)

= L.H.S.

∴ L.H.S. = R.H.S.