Sum

**Verify whether the following sequence is G.P. If so, write t _{n}:**

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

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#### Solution

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

t_{1} = `sqrt(5), "t"_2 = 1/sqrt(5), "t"_3 = 1/(5sqrt(5)), "t"_4 = 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}

∴ tn = `sqrt(5) (1/5)^("n" - 1)`

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

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

Concept: Sequence and Series - Geometric Progression (G.P.)

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