If a, b, c, d, e are in continued proportion, prove that a : e = a⁴ : b⁴.

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

Let, $\dfrac{a}{b} = \dfrac{b}{c} = \dfrac{c}{d} = \dfrac{d}{e} = k.$

∴ d = ek, c = ek², b = ek³, a = ek⁴.

Now,

$\text{L.H.S. } = \dfrac{a}{e} \\[1em] = \dfrac{ek^4}{e} = k^4 \\[1em] \text{R.H.S. } = \dfrac{(a^4)}{(b^4)} \\[1em] = \dfrac{(ek^4)^4}{(ek^3)^4} \\[1em] = \dfrac{e^4k^{16}}{e^4k^{12}} \\[1em] = k^4$

Since, L.H.S. = R.H.S. hence, proved that a : e = a⁴ : b⁴.