Permutations and Combinations

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

Question

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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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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$rightwards double arrow$ $straight x equals space fraction numerator negative 5 plus-or-minus square root of 25 minus 68 end root over denominator 2 end fraction$
$rightwards double arrow$ 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 $8 over 2 cubed comma space 8 over 2 comma space 8 left parenthesis 2 right parenthesis. space 8. space 2 cubed space space or space space space 1 comma space 4 comma space 16 comma space 64$
When a = 8, $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$ 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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Permutations And Combinations

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

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