Question

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

Answer 1

As a, b, c, are in continued proportion
Let `a/b = b/c` = k
L.H.S. = (a + b + c) (a – b + c) 
= (ck² + ck + c)(ck² – ck + c)
= c(k² + k + 1)c(k² – k + 1)
= c²(k² + k + 1)(k² – k + 1)
= c²(k⁴ + k^{2 +} 1)
R.H.S. = a² + b² + c²
= (ck²)² + (ck)² + (c)²
= c²k⁴ + c²k² + c²
= c²(k⁴ + k² + 1)
∴ L.H.S. = R.H.S.