Sponsor Area

Permutations And Combinations

Question

The product of three numbers in G.P. is 125 and the sum of their products taken in pairs is $87 1 half.$ Find them.

Solution

Let the three numbers be $space space straight a over straight r comma space straight a comma space ar$
Since their product is 125
∴    $WiredFaculty$ $rightwards double arrow$ $WiredFaculty$ $rightwards double arrow$ a = 5
Since the sum of their products taken in pairs is $87 1 half$
∴    $space space open parentheses straight a over straight r close parentheses left parenthesis straight a right parenthesis plus left parenthesis straight a right parenthesis space left parenthesis ar right parenthesis space plus space left parenthesis ar right parenthesis open parentheses straight a over straight r close parentheses space equals space 87 1 half$    $rightwards double arrow$ $straight a squared over straight r plus straight a squared straight r plus straight a squared space equals space 175 over 2$
$rightwards double arrow$ $space space space space space space space space 25 over straight r plus 25 straight r plus 25 space equals space 175 over 2$ $rightwards double arrow$    $25 plus 25 straight r squared plus 25 straight r space equals space 175 over 2 straight r$
$rightwards double arrow$    $50 straight r squared minus 125 straight r plus 50 equals 0$ $rightwards double arrow$    $2 straight r squared minus 5 straight r plus 2 equals 0$

$rightwards double arrow$    $straight r equals fraction numerator 5 plus-or-minus square root of 25 minus 16 end root over denominator 4 end fraction space equals space fraction numerator 5 plus-or-minus 3 over denominator 4 end fraction equals 2 comma space 1 half$
When a = 5 and r = 2, numbers are $5 over 2 comma space space 5 comma space space 5 left parenthesis 2 right parenthesis$ or $WiredFaculty$
When a = 5 and r = $1 half comma$ numbers are $fraction numerator 5 over denominator begin display style 1 half end style end fraction comma space 5 comma space 5 open parentheses 1 half close parentheses$ or 10, 5, $5 over 2$