Permutations and Combinations

Permutations and Combinations

Question

Find four numbers in GP. whose sum is 85 and product is 4096.

Answer

Let the four numbers in G.P. be space space straight a over straight r cubed space straight a over straight r comma space space ar comma space space ar cubed
Since their product is 4096.

∴  open parentheses straight a over straight r cubed close parentheses space open parentheses straight a over straight r close parentheses space left parenthesis ar right parenthesis space left parenthesis ar cubed right parenthesis space equals space 4096 space rightwards double arrow space space space straight a to the power of 4 space equals space 4096 space space rightwards double arrow space straight a to the power of 4 space equals space 8 to the power of 4 space rightwards double arrow space straight a space equals space 8

Since their sum is 85

∴  straight a over straight r cubed plus straight a over straight r plus ar plus ar cubed space equals space 85 space rightwards double arrow space space straight a open parentheses straight r cubed plus 1 over straight r cubed close parentheses plus straight a open parentheses straight r plus 1 over straight r close parentheses space equals space 85
rightwards double arrow   8 open square brackets open parentheses straight r plus 1 third close parentheses cubed minus space 3 straight r 1 over straight r open parentheses straight r plus 1 over straight r close parentheses close square brackets plus 8 open parentheses straight r plus 1 over straight r close parentheses space equals space 85
Put                      straight r plus 1 over straight r space equals straight t
∴      8 straight t cubed minus 24 straight t plus 8 straight t space equals space 85 space space space space space space or space space space space space 8 straight t cubed minus 16 straight t minus 85 space equals space 0  or  
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#5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array)
#6 {main}</pre>
Putting                               2t = x, we get
    straight x cubed minus 8 straight x minus 85 equals 0
By inspection x = 5 satisfies the equation. So x - 5 is a factor of straight x cubed minus 8 straight x minus 85. Other roots are given by 
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#6 {main}</pre>
rightwards double arrow                  straight x equals space fraction numerator negative 5 plus-or-minus square root of 25 minus 68 end root over denominator 2 end fraction   
rightwards double arrow    x is imaginary
∴      x = 5 is the only real value
So,                       2t = 5,
space space space rightwards double arrow space space space space straight t space equals space 5 over 2 comma space space straight r plus space 1 over straight r space equals space 5 over 2
space space space rightwards double arrow space space space space  2 straight r squared minus 5 straight r plus 2 space equals space 0
                       straight r equals space 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 space equals space 2 comma space space 1 half
When a = 8, r = 2, numbers are 8 over 2 cubed comma space 8 over 2 comma space 8 left parenthesis 2 right parenthesis. space 8. space 2 cubed space space or space space space 1 comma space 4 comma space 16 comma space 64
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#6 {main}</pre> numbers are 8 over open parentheses begin display style 1 half end style close parentheses cubed comma space fraction numerator 8 over denominator begin display style 1 half end style end fraction comma space 8 open parentheses 1 half close parentheses comma space 8 open parentheses 1 half close parentheses cubed  or 64, 16, 4, 1


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