Question

If a,b,c are in continued proportion, prove that (a2+b2+c2)(b2+c2+d2)=(ab+bc+cd)2​

Answer 1

Answer:

Solution :-

a/b = b/c = c/d = k (let)

a = bk

b = ck

c = dk

L.H.S :-

★ (a² + b² + c²) (b² + c² + d²)

➻ (b²k² + c²k² + d²k²) (c²k² + d²k² + d²)

➻ (c²k⁴ + c²k² + d²k²) (d²k²k² + d²k² + d²)

➻ (d²k⁶ + d²k⁴ + d²k²) (d²k⁴ + d²k² + d²)

➻ d²k²(k⁴ + k² + 1) d²(k⁴ + k² + 1)

➻ d⁴k²(k⁴ + k² + 1)²

R.H.S :-

★ (ab + bc + cd)²

➻ (bk.b + ck.c + dk.d)²

➻ (b²k + c²k + d²k)²

➻ (c²k²k + d²k²k + d²k)²

➻ (d²k²k² × k + d²k³ + d²k)²

➻ (d²k⁵ + d²k³ + d²k)²

➻ {d²k(k⁴ + k² + 1)}²

➻ d⁴k²(k⁴ + k² + 1)²

∴ L.H.S = R.H.S (Proved)