Permutations and Combinations

Permutations and Combinations

Question

Find the sum of the squares of first n natural numbers. Or

Evaluate  space space sum straight n squared space space space or space space space space sum from straight k equals 1 to straight n of space straight k squared

Answer

Consider an identity left parenthesis straight x plus 1 right parenthesis cubed minus space straight x cubed space equals space 3 straight x squared plus 3 straight x plus 1
Substituting x = 1, 2, 3,........n, we get <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)
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                      4 cubed minus 3 cubed space equals space 3.3 squared plus 3.3 plus 1
                       ............................
                       .............................
           left parenthesis straight n plus 1 right parenthesis cubed minus straight n cubed space equals space 3. straight n squared plus 3. straight n plus 1
Adding vertically, we get left parenthesis straight n plus 1 right parenthesis cubed minus 1 cubed space equals space 3 left parenthesis 1 squared plus 2 squared plus 3 squared plus....... plus straight n squared right parenthesis space plus 3 left parenthesis 1 plus 2 plus 3 plus..... plus straight n right parenthesis space plus space straight n

rightwards double arrow  straight n cubed plus 3 straight n squared plus 3 straight n plus 1 minus 1 space equals space 3 sum straight n squared plus 3 sum straight n plus straight n

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#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)
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#6 {main}</pre>
                                                    equals straight n open square brackets straight n squared plus 3 straight n plus 2 minus open parentheses fraction numerator 3 straight n plus 3 over denominator 2 end fraction close parentheses close square brackets
rightwards double arrow                     sum straight n squared space equals space straight n over 3 open square brackets fraction numerator 2 straight n squared plus 6 straight n plus 4 minus 3 straight n minus 3 over denominator 2 end fraction close square brackets space equals space fraction numerator straight n left parenthesis 2 straight n squared plus 3 straight n plus 1 right parenthesis over denominator 6 end fraction space equals space fraction numerator straight n left parenthesis 2 straight n squared plus 2 straight n plus straight n plus 1 right parenthesis over denominator 6 end fraction
                                 equals fraction numerator straight n left square bracket 2 straight n left parenthesis straight n plus 1 right parenthesis plus 1 left parenthesis straight n plus 1 right parenthesis over denominator 6 end fraction space equals space fraction numerator straight n left parenthesis straight n plus 1 right parenthesis left parenthesis 2 straight n plus 1 right parenthesis over denominator 6 end fraction
∴   space space space space space space sum from straight k equals 1 to straight n of space straight k squared  or  sum straight n squared space equals space 1 squared plus 2 squared plus 3 squared plus....... plus straight n squared space equals space fraction numerator straight n left parenthesis straight n plus 1 right parenthesis left parenthesis 2 straight n plus 1 right parenthesis over denominator 6 end fraction
                          

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