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Question
Find the value of n so that `(a^(n+1) + b^(n+1))/(a^n + b^n)` may be the geometric mean between a and b.
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Solution
The geometric mean between a and b = `sqrt"ab"`
⇒ `("a"^("n"+ 1) + "b"^("n" + 1))/("a"^"n" + "b"^"n") = sqrt"ab"`
∴ `"a"^("n"+ 1) + "b"^("n" + 1) = sqrt"ab" ("a"^"n" + "b"^"n")`
= `"a"^("n"+ 1/2) "b"^(1/2) + "a"^(1/2) "b"^("n" + 1/2)`
or `("a"^("n" + 1) - "a"^("n" + 1/2) "b"^(1/2)) - ("a"^(1/2) "b"^("n" + 1/2) - "b"^("n" + 1)) = 0`
or `"a"^("n" + 1/2) ("a"^(1/2) - "b"^(1/2)) - "b"^ ("n" + 1/2)("a" ^(1/2) - "b"^(1/2)) = 0`
or `("a"^(1/2) - "b"^(1/2)) ("a"^("n" + 1/2) - "b"^ ("n" + 1/2)) = 0`
`"a" ^(1/2) - "b"^(1/2) ≠ 0`
∴ `"a"^("n" + 1/2) - "b"^ ("n" + 1/2) = 0`
or `"a"^("n" + 1/2) = "b"^ ("n" + 1/2)`
or `("a"/"b")^("n"+1/2) = 1 = ("a"/"b")^0`
⇒ `"n"+ 1/2 = 0`
n = `(-1)/2`
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