Sets

Question 1
CBSEENMA11012915

Use principle of mathematical induction to prove that:

1 space plus space 2 space plus space 3 space plus space... space plus space straight n space equals space fraction numerator straight n left parenthesis straight n space plus space 1 right parenthesis over denominator 2 end fraction

Solution

Let P(n): 1 + 2 + 3 + ......... + n = space space fraction numerator straight n left parenthesis straight n plus 1 right parenthesis over denominator 2 end fraction
I. For n = 1,
    P(1) : 1 = fraction numerator 1 left parenthesis 1 plus 1 right parenthesis over denominator 2 end fraction rightwards double arrow space 1 space equals space 1 space rightwards double arrow space space straight P left parenthesis 1 right parenthesis is true.
II.  Suppose the statement is true for n = m, straight m element of straight N
      i.e. P(m): 1 plus 2 plus 3 plus........ space plus straight m space equals space fraction numerator straight m left parenthesis straight m plus 1 right parenthesis over denominator 2 end fraction          ....(i)
III.    For n = m + 1,
        P(m + 1): 1 + 2 + 3 + ........ + (m + 1) = fraction numerator left parenthesis straight m plus 1 right parenthesis left parenthesis straight m plus 2 right parenthesis over denominator 2 end fraction
or  [1 + 2 + 3 + ...... + m] + (m + 1) = fraction numerator left parenthesis straight m plus 1 right parenthesis left parenthesis straight m plus 2 right parenthesis over denominator 2 end fraction
                                       
                                         [From (i), 1 + 2 + 3 + ...... + m = fraction numerator straight m left parenthesis straight m plus 1 right parenthesis over denominator 2 end fraction]
∴        P (m + 1): space fraction numerator straight m left parenthesis straight m space plus space 1 right parenthesis over denominator 2 end fraction space plus space left parenthesis straight m space plus space 1 right parenthesis space equals space fraction numerator left parenthesis straight m plus 1 right parenthesis left parenthesis straight m plus 2 right parenthesis over denominator 2 end fraction
rightwards double arrowspace space left parenthesis straight m plus 1 right parenthesis open parentheses straight m over 2 plus 1 close parentheses space equals space fraction numerator left parenthesis straight m plus 1 right parenthesis left parenthesis straight m plus 2 right parenthesis over denominator 2 end fraction
rightwards double arrow   left parenthesis straight m plus 1 right parenthesis open parentheses fraction numerator straight m plus 2 over denominator 2 end fraction close parentheses space equals space fraction numerator left parenthesis straight m plus 1 right parenthesis space left parenthesis straight m plus 2 right parenthesis over denominator 2 end fraction
rightwards double arrowfraction numerator left parenthesis straight m plus 1 right parenthesis left parenthesis straight m plus 2 right parenthesis over denominator 2 end fraction space equals space fraction numerator left parenthesis straight m plus 1 right parenthesis left parenthesis straight m plus 2 right parenthesis over denominator 2 end fraction
    which is true

∴    P(m + 1) is true

∴    P(m) is true rightwards double arrow P(m + 1) is true
Hence, by mathematical induction
P(n) is true for all space space straight n element of straight N.


      

Question 2
CBSEENMA11012916

Prove the following by using the principle of mathematical induction for all straight n space element of space straight N colon

1 cubed plus 2 cubed plus 3 cubed plus space... space plus space straight n cubed space equals space open square brackets fraction numerator straight n left parenthesis straight n plus 1 right parenthesis over denominator 2 end fraction close square brackets squared

Solution

Let straight P left parenthesis straight n right parenthesis space colon space 1 cubed plus 2 cubed plus 3 cubed plus..... plus straight n cubed space equals space open square brackets fraction numerator straight n left parenthesis straight n plus 1 right parenthesis over denominator 2 end fraction close square brackets squared
I.      For n = 1,
       straight P left parenthesis 1 right parenthesis colon space 1 cubed space equals space open square brackets fraction numerator 1 left parenthesis 1 plus 1 right parenthesis over denominator 2 end fraction close square brackets squared space rightwards double arrow 1 space equals space 1 space rightwards double arrow space straight P left parenthesis 1 right parenthesis is true.
II.    Suppose the statement is true for n = m, straight m space element of space straight N
   
          i.e., "<pre    ... (i)
III.     For n = m + 1,
        straight P left parenthesis straight m plus 1 right parenthesis colon space 1 cubed plus 2 cubed plus 3 cubed plus......... plus left parenthesis straight m plus 1 right parenthesis cubed space equals space open square brackets fraction numerator left parenthesis straight m plus 1 right parenthesis left parenthesis straight m plus 2 right parenthesis over denominator 2 end fraction close square brackets squared
or     left square bracket 1 cubed space plus space 2 cubed space plus space 3 cubed space plus space....... space plus space straight m cubed right square bracket space plus space left parenthesis straight m plus 1 right parenthesis cubed space equals space open square brackets fraction numerator left parenthesis straight m plus 1 right parenthesis left parenthesis straight m plus 2 right parenthesis over denominator 2 end fraction close square brackets squared
     From (i), 1 cubed plus 2 cubed plus 3 cubed plus space............ space plus space straight m cubed space equals space open square brackets fraction numerator straight m left parenthesis straight m plus 1 right parenthesis over denominator 2 end fraction close square brackets squared

∴   space space straight P left parenthesis straight m plus 1 right parenthesis space colon space open square brackets fraction numerator straight m left parenthesis straight m plus 1 right parenthesis over denominator 2 end fraction close square brackets squared space plus space left parenthesis straight m plus 1 right parenthesis cubed space equals space open square brackets fraction numerator left parenthesis straight m plus 1 right parenthesis left parenthesis straight m plus 2 right parenthesis over denominator 2 end fraction close square brackets squared
rightwards double arrow space space left parenthesis straight m plus 1 right parenthesis squared space open square brackets straight m squared over 4 plus left parenthesis straight m plus 1 right parenthesis close square brackets space equals space open square brackets fraction numerator left parenthesis straight m plus 1 right parenthesis left parenthesis straight m plus 2 right parenthesis over denominator 2 end fraction close square brackets squared
rightwards double arrow space space space left parenthesis straight m plus 1 right parenthesis squared open parentheses fraction numerator straight m squared plus 4 straight m plus 4 over denominator 4 end fraction close parentheses space equals space open square brackets fraction numerator left parenthesis straight m plus 1 right parenthesis left parenthesis straight m plus 2 right parenthesis over denominator 2 end fraction close square brackets squared
rightwards double arrow space space space fraction numerator left parenthesis straight m plus 1 right parenthesis squared left parenthesis straight m plus 2 right parenthesis squared over denominator 4 end fraction space equals space open square brackets fraction numerator left parenthesis straight m plus 1 right parenthesis left parenthesis straight m plus 2 right parenthesis over denominator 2 end fraction close square brackets squared space rightwards double arrow space open square brackets fraction numerator left parenthesis straight m plus 1 right parenthesis left parenthesis straight m plus 2 right parenthesis over denominator 2 end fraction close square brackets squared space equals space open square brackets fraction numerator left parenthesis straight m plus 1 right parenthesis space left parenthesis straight m plus 2 right parenthesis over denominator 2 end fraction close square brackets squared
       which is true

∴     P(m + 1) is true

∴     P(m) is true rightwards double arrow P(m + 1) is true.
Hence, by mathematical induction, P(n) is true for all straight n element of space straight N.



Question 3
CBSEENMA11012917

Prove the following by using the principle of mathematical induction for all straight n element of straight N:

1 plus 3 plus 3 squared plus....... space plus 3 to the power of straight n minus 1 end exponent space equals space fraction numerator 3 to the power of straight n minus 1 over denominator 2 end fraction

Solution

 Let P(n) : <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>
I.    For n = 1,
      P(1) : 1 equals fraction numerator 3 to the power of 1 minus 1 over denominator 2 end fraction space rightwards double arrow space 1 space equals space 2 over 2 space rightwards double arrow space space space 1 space equals space 1

∴     P(1) is true.
II.   Let the statement be true for n = m,  straight m element of space straight N

∴     P(m) : 1 plus 3 plus 3 squared plus space........... space plus space 3 to the power of straight m minus 1 end exponent space equals space fraction numerator 3 to the power of straight m minus 1 over denominator 2 end fraction     ... (i)

III.  
For   n = m + 1,                      
      P(m + 1) : <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    straight P left parenthesis straight m plus 1 right parenthesis space colon space 1 plus 3 plus 3 squared plus.......... plus 3 to the power of straight m minus 1 end exponent space plus 3 to the power of straight m equals space fraction numerator 3 to the power of straight m minus 1 end exponent minus 1 over denominator 2 end fraction
  From (i), <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>

∴    straight P left parenthesis straight m space plus 1 right parenthesis space colon space open parentheses fraction numerator 3 to the power of straight m minus 1 over denominator 2 end fraction close parentheses space plus space 3 to the power of straight m space equals space fraction numerator 3 to the power of straight m plus 1 end exponent minus 1 over denominator 2 end fraction
open parentheses fraction numerator 3 to the power of straight m minus 1 space plus space 2. space 3 to the power of straight m over denominator 2 end fraction close parentheses space equals space fraction numerator 3 to the power of straight m plus 1 end exponent minus 1 over denominator 2 end fraction rightwards double arrow space space fraction numerator 3.3 to the power of straight m plus 1 end exponent minus 1 over denominator 2 end fraction space equals space fraction numerator 3 to the power of straight m plus 1 end exponent minus 1 over denominator 2 end fraction space space rightwards double arrow space fraction numerator 3 to the power of straight m plus 1 end exponent minus 1 over denominator 2 end fraction space equals space fraction numerator 3 to the power of straight m plus 1 end exponent minus 1 over denominator 2 end fraction
which is true.

∴   P(m + 1) is true.

∴   P(m) is true rightwards double arrowP (m + 1) is true
Hence, by principle of mathematical induction, P(n) is true for all straight n element of space straight N.




Question 4
CBSEENMA11012918

Prove the following by using the principle of mathematical induction for all straight n element of straight N:

<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>

Solution

Let P(n):  1 half plus space 1 fourth plus 1 over 8 plus....... plus 1 over 2 to the power of straight n space equals space 1 minus space 1 over 2 to the power of straight n
I.    For n = 1,
     P(1) :  1 half space equals space 1 space minus space 1 over 2 to the power of 1 rightwards double arrow space 1 half space equals space 1 space minus space 1 half space space rightwards double arrow space 1 half space equals space 1 half
∴    P(1) is true.
II.   Let the statement be true
       for n = m, straight m space element of space straight N

∴   <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>       .... (i)

III.  
For n = m + 1,
       P(m + 1): 1 half plus 1 fourth plus 1 over 8 plus......... plus 1 over 2 to the power of straight m plus 1 end exponent space equals space 1 space minus space 1 over 2 to the power of straight m plus 1 end exponent
or     space space 1 half space plus space 1 fourth space plus space 1 over 8 space plus space......... space plus space 1 over 2 to the power of straight m plus 1 over 2 to the power of straight m plus 1 end exponent space equals space 1 space minus 1 over 2 to the power of straight m plus 1 end exponent
        From (i),
       1 half space plus space 1 fourth space plus space 1 over 8 space plus space.......... space plus space 1 over 2 to the power of straight m space equals space 1 space minus space 1 over 2 to the power of straight m

∴   straight P left parenthesis straight m plus 1 right parenthesis colon space 1 space minus space 1 over 2 to the power of straight m space plus space 1 over 2 to the power of straight m plus 1 end exponent space equals space 1 space minus space 1 over 2 to the power of straight m plus 1 end exponent
 rightwards double arrow space 1 minus open parentheses 1 over 2 to the power of straight m minus 1 over 2 to the power of straight m plus 1 end exponent close parentheses space equals space 1 space minus space 1 over 2 to the power of straight m plus 1 end exponent space rightwards double arrow space 1 minus space open parentheses fraction numerator 2 minus 1 over denominator 2 to the power of straight m plus 1 end exponent end fraction close parentheses space equals space 1 minus space 1 over 2 to the power of straight m plus 1 end exponent space
rightwards double arrow space 1 minus 1 over 2 to the power of straight m plus 1 end exponent space equals space 1 minus space 1 over 2 to the power of straight m plus 1 end exponent
      which is true.
∴    P(m + 1) is true
∴    P(m) is true rightwards double arrow P(m + 1) is true
Hence, by the principal of mathematical induction, P(n) is true for all straight n element of straight N.


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