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Question 27

Find the value of n so that fraction numerator a to the power of n plus 1 end exponent space plus space b to the power of n plus 1 end exponent over denominator a to the power of n space plus space b to the power of n end fraction may be the geometric mean between a and b.

Answer

G. M. of a and b is square root of a b end root.

By the given condition, fraction numerator a to the power of n plus 1 end exponent space plus space b to the power of n plus 1 end exponent over denominator a to the power of n space plus space b to the power of n end fraction equals square root of a b end root

Squaring both sides, we obtain

open parentheses a to the power of n plus 1 end exponent space plus space b to the power of n plus 1 end exponent close parentheses squared over open parentheses a to the power of n space plus space b to the power of n close parentheses squared equals a b rightwards double arrow a to the power of 2 n plus 2 end exponent space plus space 2 a to the power of n plus 1 end exponent b to the power of n plus 1 end exponent space plus space b to the power of 2 n plus 2 end exponent space equals space open parentheses a b close parentheses open parentheses a to the power of 2 n end exponent space plus space 2 a to the power of n b to the power of n space plus space b to the power of 2 n end exponent close parentheses rightwards double arrow a to the power of 2 n plus 2 end exponent space plus space 2 a to the power of n plus 1 end exponent b to the power of n plus 1 end exponent space plus space b to the power of 2 n plus 2 end exponent space equals space a to the power of 2 n plus 1 end exponent b space plus space 2 a to the power of n plus 1 end exponent b to the power of n plus 1 end exponent space plus space a b to the power of 2 n plus 1 end exponent rightwards double arrow a to the power of 2 n plus 2 end exponent space plus space b to the power of 2 n plus 2 end exponent space equals space a to the power of 2 n plus 1 end exponent b space plus space a b to the power of 2 n plus 1 end exponent rightwards double arrow a to the power of 2 n plus 2 end exponent space minus a to the power of 2 n plus 1 end exponent b space equals space a b to the power of 2 n plus 1 end exponent space minus space b to the power of 2 n plus 2 end exponent rightwards double arrow space a to the power of 2 n plus 1 end exponent open parentheses a minus b close parentheses equals b to the power of 2 n plus 1 end exponent open parentheses a minus b close parentheses rightwards double arrow open parentheses a over b close parentheses to the power of 2 n plus 1 end exponent space equals space 1 space equals space open parentheses a over b close parentheses to the power of 0 rightwards double arrow 2 n space plus 1 space equals space 0 rightwards double arrow n space equals space fraction numerator minus 1 over denominator 2 end fraction

 

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