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Question
If the first and the nth term of a G.P. are a ad b, respectively, and if P is the product of n terms, prove that P2 = (ab)n.
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Solution
Let the geometric series be a common ratio.
First term = a, nth term = ar n – 1 = b
P = product of n terms
= a. ar. ar2. ar3 …. arn – 1
= `"a"^"n". "r"^ (1+ 2 + 3 + ... +("n" - 1))`
= `"a"^"n""r"^(("n"("n" - 1))/2)`
p2 = `"a"^(2"n") "r"^("n"("n" - 1))` ..........(i)
(ab)n = `("a" xx "ar"^("n" - 1))^"n"`
= `("a"^2 xx "r"^("n" - 1))^"n"`
= `"a"^(2"n"). "r"^("n"("n" - 1))` ..........(ii)
From equations (i) and (ii),
P2 = (ab)n
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