Question

If a, b, c, d are in continued proportion, prove that: (a² – b²) (c² – d²) = (b² – c²)² 

Answer 1

a, b, c, d are in continued proportion
∴ `a/b = b/c = c/d` = k(say)
∴ c = dk, b = ck = dk. k = dk²,
a = bk = dk². k = dk³
L.H.S. = (a² – b²) (c² – d²)
= [(dk³)² – (dk²)²] [(dk)² – d²]
= (d²k⁶ – d²k⁴) (d²k² – d²)
= d²k⁴ (k² – 1) d²(k² – 1)
= d⁴k⁴ (k² – 1)²
R.H.S. = (b² – c²)²
= [(dk²)² – (dk)²]²
= [d²k² – d²k²]²
= [d²k² (k² – 1)]²
= d⁴k⁴(k² – 1)²
∴ L.H.S. = R.H.S.