Permutations And Combinations

Question
CBSEENMA11013154

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

Solution

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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#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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