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Question
Answer the following:
For a sequence , if tn = `(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.
Solution
tn = `(5^("n" - 2))/(7^("n" - 3)) = (5.5^("n" - 3))/(7^("n" - 3))`
∴ tn = `5(5/7)^("n" - 3)`
∴ tn+1 = `5(5/7)^("n" + 1 - 3)`
= `5(5/7)^("n" - 2)`
∴ `("t"_("n" + 1))/"t"_"n" = (5(5/7)^("n" - 2))/(5(5/7)^("n" - 3))`
= `(5/7)^("n" - 2 - "n" + 3)`
= `5/7`, which is a constant
∴ the sequence is a G.P. whose common ratio is `5/7`
Now, tn = `5(5/7)^("n" - 3)`
∴ the first term = t1 = `5(5/7)^(1 - 3)`
= `5(5/7)^(-2)`
= `5(7/5)^2`
= `49/5`
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