Question

Verify whether the following sequence is G.P. If so, find t_n.

`sqrt(5), 1/sqrt(5), 1/(5sqrt(5)), 1/(25sqrt(5)), ...`

Answer 1

`sqrt(5), 1/sqrt(5), 1/(5sqrt(5)), 1/(25sqrt(5)), ...`

t₁ = `sqrt(5)`, t₂ = `1/sqrt(5)`, t₃ = `1/(5sqrt(5))`, t₄ = `1/(25sqrt(5)`, ...

Here, `"t"_2/"t"_1 = "t"_3/"t"_2 = "t"_4/"t"_3 = 1/5`

Since, the ratio of any two consecutive terms is a constant, the given sequence is a geometric progression.

Here, a = `sqrt(5)`, r = `1/5`

t_n = ar^n–1

∴ t_n = `sqrt(5) (1/5)^("n" - 1)`

= `(5)^(1/2) (5)^(1 - "n")`

= `(5)^(3/2 - "n")`