If a, b, c and d are in continued proportion, prove that ad(c 2 + d 2 ) = c 3 (b + d).

Since, a, b, c, d are in continued proportion.

$\therefore \frac{a}{b} = \frac{b}{c} = \frac{c}{d} = k$ (let).

c = dk, b = ck = (dk)k = dk², a = bk = (dk²)k = dk³.

Substituting values in L.H.S. of the equation ad(c² + d²) = c³(b + d), we get :

L.H.S = ad(c² + d²)

= dk³.(d).[(dk)² + d²]

= d²k³.[d²(k² + 1)]

= d⁴k³(k² + 1).

Substituting values in R.H.S. of the equation ad(c² + d²) = c³(b + d), we get :

R.H.S = c³(b + d)

= (dk)³.(dk² + d)

= d³k³[d(k² + 1)]

= d⁴k³(k² + 1).

Since, L.H.S = R.H.S

Hence, proved that ad(c² + d²) = c³(b + d).