Question

If t_k is the k^th term of a G.P., then show that t_{n – k}, t_n, t_{n + k} also form a GP for any positive integer k

Answer 1

Let a be the first term and r be the common ratio.

We are given t_k = ar^{k – 1}

We have to prove: t_{n – k}, t_n, t_{n + k} form a G.P.

t_{n – k} = `"ar"^("n" - "k" - 1)`

t_n = ar^{n – 1} 

t_{n + k} = `"ar"^("n" + "k" - 1)`

Now `"t"_"n"/"t"_("n" - "k") = ("ar"^("n" - 1))/("ar"^("n" - "k" - 1))`

= `"r"^("n"-1 - "n" + "k" + 1)`

= r^k

Also `("t"_("n" + "k"))/"t"_"n" =("ar"^("n" + "k" - 1))/("ar"^("n" - 1))`

= `"r"^("" + "k - 1 - "n" + 1)`

= r^k

Now `"t"_"n"/("t"_("n" -"k")) = ("t"_("n"+ "k"))/"t"_"n"`

⇒ t_{n – k}, t_n, t_{n + k} fom a G.P.