Permutations and Combinations

Permutations and Combinations

Question

How many terms of the A.P. -6, fraction numerator negative 11 over denominator 2 end fraction comma -5, .......... are needed to give the sum -25? Explain the double answer.

Answer

The given A.P. is: -6, fraction numerator negative 11 over denominator 2 end fraction comma -5, .............
Here, a = -6,  straight d equals fraction numerator negative 11 over denominator 2 end fraction minus left parenthesis negative 6 right parenthesis space equals space fraction numerator negative 11 over denominator 2 end fraction space plus space 6 space equals space 1 half
Let -25 be the sum of n terms

∴           straight S subscript straight n space equals space minus 25   or    space space straight n over 2 left square bracket 2 straight a plus left parenthesis straight n minus 1 right parenthesis straight d right square bracket space equals space minus 25
or    straight n open square brackets 2 left parenthesis negative 6 right parenthesis space plus space left parenthesis straight n minus 1 right parenthesis 1 half close square brackets space equals space minus 50    or      straight n open square brackets negative 12 space plus space fraction numerator straight n minus 1 over denominator 2 end fraction close square brackets space equals space minus 50
or            <pre>uncaught exception: <b>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</b><br /><br />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<br />#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>            or                   n(n - 25) = -100
or             straight n squared minus 25 straight n plus 100 equals 0
or                               space space space straight n equals fraction numerator 25 plus-or-minus square root of left parenthesis 25 right parenthesis squared minus 4.1.100 end root over denominator 2.1 end fraction space equals space fraction numerator 25 plus-or-minus square root of 625 minus 400 end root over denominator 2 end fraction space equals space fraction numerator 25 plus-or-minus 15 over denominator 2 end fraction equals 20.5

Here, the common difference is positive.

∴ The A.P. starts from negative terms and its terms are increasing.

∴ All the terms after 13th term are positive.

These positive terms from 14th term to 20th term when added to negative terms from 6th term to 12th term, they cancel out each other and the sum remains same.

Hence, the sum of first five terms is same as the sum of first 20 terms.

More Chapters from Permutations and Combinations