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Common Ratio

The sum of th...

Question

The sum of three numbers in G.P. is $\frac{39}{10}$ and their product is 1. Find the numbers.

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Solution

Let the number be
$a over r comma a space a n d space a r a over r cross times r a r equals 1 a cubed equals 1 a equals 1 N o w a over r plus a plus a r equals 39 over 10 1 over r plus 1 plus r equals 39 over 10 fraction numerator 1 plus r plus r squared over denominator r end fraction equals 39 over 10 10 plus 10 r plus 10 r squared equals 39 r 10 r squared minus 29 r plus 10 equals 0 10 r squared minus 25 r minus 4 r plus 10 equals 0 5 r left parenthesis 2 r minus 5 right parenthesis minus 2 left parenthesis 2 r minus 5 right parenthesis left parenthesis 2 r minus 5 right parenthesis left parenthesis 5 r minus 2 right parenthesis equals 0 r equals 5 over 2 space space o r space space space r equals 2 over 5 t h u s space comma r e q u i r e d space t e r m s space a r e colon a over r comma a. a r equals fraction numerator 1 over denominator begin display style 5 over 2 end style end fraction comma 1 comma 1 cross times 5 over 2 space space space space o r space space space fraction numerator 1 over denominator begin display style 2 over 5 end style end fraction comma 1 comma 1 cross times 2 over 5 equals 2 over 5 comma 1 comma 5 over 2 space space space space space o r space space space space 5 over 2 comma 1 comma 2 over 5$

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