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Question
Check whether the following sequence is G.P. If so, write tn.
`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))`, ...
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`
∴ the ratio of any two consecutive terms is a constant, hence the given sequence is a Geometric progression.
Here, a = `sqrt(5)`, r = `1/5`,
tn = arn–1
∴ tn = `sqrt(5)(1/5)^("n" - 1)`
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