Question

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

Answer 1

Let a₁, a₂, a₃, ................., a_n, .......... be a G.P. with common ratio r.

`=> (a_(n + 1))/a_n = r` for all n ∈ N

If each term of a G.P. is raised to the power x, we get the sequence `a_1^x, a_2^x, a_3^x, ............, a_n^x,.........`

Now, `(a_(n + 1))^x/(a_n)^x = ((a_(n + 1))/a_n)^x = r^x` for all n ∈ N

Hence, `a_1^x, a_2^x, a_3^x, ............, a_n^x,.........` is also a G.P.