If a, b, c and d are in G.P. show that (a² + b² + c²) (b² + c² + d²) = (ab + bc + cd)².

Question

If a, b, c and d are in G.P. show that (a 2 + b 2 + c 2 ) (b 2 + c 2 + d 2 ) = (ab + bc + cd) 2 .

Philoid · Accepted Answer

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

Here,

a, b, c, d are in G.P.

∴ bc = ad — (1)

b² = ac —(2)

c² = bd –(3)

we have to be prove that,

(a2 + b2 + c2) (b2 + c2 + d2) = (ab + bc – cd)2

From R.H.S.

(ab + bc + cd)²

= (ab + ad + cd)² (From eq–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 (from eq– 1 and eq—2)

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

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

From eq – 2 and eq – 3

= 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.

∴(a² + b² + c²) (b² + c² + d²) = (ab + bc + cd)².

If a, b, c and d are in G.P. show that (a2 + b2 + c2) (b2 + c2 + d2) = (ab + bc + cd)2.