If each term of a G.P. is raised to the power x, show that the resulting sequence is also a G.P.

Let a, b and c be in G.P.

∴ b² = ac

Let, b² = ac = m ……..(i)

Each term be raised to power x.

So, terms = a^x, b^x and c^x.

Terms are in G.P. if common ratio between terms are equal,

$\Rightarrow \dfrac{b^x}{a^x} = \dfrac{c^x}{b^x} \\[1em] \Rightarrow b^x.b^x = c^x.a^x \\[1em] \Rightarrow (b^x)^2 = (a^x.c^x) \\[1em] \Rightarrow (b^2)^x = (ac)^x \\[1em] \text{From (i)},\\[1em] \Rightarrow m^x = m^x.$

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

Hence, proved that on raising each term of a G.P. to the power x, the resulting sequence is also a G.P.