Question

For a sequence, if t_n = `(5^("n" - 2))/(7^("n" - 3))`, verify whether the sequence is a G.P. If it is a G.P., find its  first term and the common ratio.

Answer 1

The sequence (t_n) is a G.P. if 

`("t"_("n" + 1))/("t"_"n")` = constant for all n ∈ N.

Now, t_n = `(5^("n" - 2))/(7^("n" - 3))`

∴ t_n+1 = `(5^("n" + 1- 2))/(7^("n" + 1 - 3)) = (5^("n" - 1))/(7^("n" - 2))`

∴ `("t"_("n" + 1))/"t"_"n" = (5^("n" - 1))/(7^("n" - 2)) = (7^("n" - 3))/(5^("n" - 2))`

= `5^(("n" - 1) - ("n" - 2)) xx 7^(("n" - 3) - ("n" - 2)`

= 5⁽¹⁾ x 7^–1 = `5/7`

= constant , for all n ∈ N.

∴ the sequence is a G.P. with common ratio (r) = `5/7` 

and first term = t₁ = `(5^(1 - 2))/(7^(1 - 3))`

= `(5^(-1))/(7^(-2)) = 7^2/5 = 49/5`.