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

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

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 $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$

 $4 cubed minus 3 cubed space equals space 3.3 squared plus 3.3 plus 1$

............................

.............................

$left parenthesis straight n plus 1 right parenthesis cubed minus straight n cubed space equals space 3. straight n squared plus 3. straight n plus 1$

Adding vertically, we get $left parenthesis straight n plus 1 right parenthesis cubed minus 1 cubed space equals space 3 left parenthesis 1 squared plus 2 squared plus 3 squared plus....... plus straight n squared right parenthesis space plus 3 left parenthesis 1 plus 2 plus 3 plus..... plus straight n right parenthesis space plus space straight n$

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

$rightwards double arrow$ $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$

 $equals straight n open square brackets straight n squared plus 3 straight n plus 2 minus open parentheses fraction numerator 3 straight n plus 3 over denominator 2 end fraction close parentheses close square brackets$

$rightwards double arrow$ $sum straight n squared space equals space straight n over 3 open square brackets fraction numerator 2 straight n squared plus 6 straight n plus 4 minus 3 straight n minus 3 over denominator 2 end fraction close square brackets space equals space fraction numerator straight n left parenthesis 2 straight n squared plus 3 straight n plus 1 right parenthesis over denominator 6 end fraction space equals space fraction numerator straight n left parenthesis 2 straight n squared plus 2 straight n plus straight n plus 1 right parenthesis over denominator 6 end fraction$

$equals fraction numerator straight n left square bracket 2 straight n left parenthesis straight n plus 1 right parenthesis plus 1 left parenthesis straight n plus 1 right parenthesis over denominator 6 end fraction space equals space fraction numerator straight n left parenthesis straight n plus 1 right parenthesis left parenthesis 2 straight n plus 1 right parenthesis over denominator 6 end fraction$

∴ $space space space space space space sum from straight k equals 1 to straight n of space straight k squared$ or $sum straight n squared space equals space 1 squared plus 2 squared plus 3 squared plus....... plus straight n squared space equals space fraction numerator straight n left parenthesis straight n plus 1 right parenthesis left parenthesis 2 straight n plus 1 right parenthesis over denominator 6 end fraction$

129 Views

Long Answer Type

122. Find the sum of the cubes of first n natural numbers. Or

Evaluate

space space sum straight n cubed space space or space space space sum from straight k space equals 1 to straight n of space straight k cubed

Also, show that

sum straight n cubed space equals space left parenthesis sum straight n right parenthesis squared

110 Views

Short Answer Type

123. Find the sum of n terms of the series whose nth term is

space space fraction numerator 1 cubed plus 2 cubed plus 3 cubed plus...... plus straight n cubed over denominator straight n end fraction.

128 Views

Long Answer Type

124.

Find the sum to n terms of the series:

1 . 2 . 4 + 2. 3 . 7 + 3. 4. 10 +..............

161 Views

Short Answer Type

125. Find the sum of the cubes of first n odd natural numbers. Or Sum the series to n terms l³ + 3³ + 5³ +..........

90 Views

126.

Find the sum to n terms the series $fraction numerator 1 over denominator 1.2 end fraction plus fraction numerator 1 over denominator 2.3 end fraction plus fraction numerator 1 over denominator 3.4 end fraction plus...........$ and deduce the sum to infinity.

102 Views

127.

Sum of the series n . 1 + (n - 1). 2 + (n - 2) . 3 + ........ + 1. n

94 Views

Long Answer Type

128. Show that the sum of the cubes of any number of consecutive integers is divisible by the sum of these integers.

154 Views

Short Answer Type

129.

Show that $fraction numerator 1 cross times 2 squared plus 2 cross times 3 squared plus...... plus straight n cross times left parenthesis straight n plus 1 right parenthesis squared over denominator 1 squared cross times 2 plus 2 squared cross times 3 plus...... plus straight n squared cross times left parenthesis straight n plus 1 right parenthesis end fraction space equals space fraction numerator 3 straight n plus 5 over denominator 3 straight n plus 1 end fraction$