Prove the following by using the principle of mathematical induc
    • Zigya Logo

    Previous Year Papers

    Download Solved Question Papers Free for Offline Practice and view Solutions Online.

    Test Series

    Take Zigya Full and Sectional Test Series. Time it out for real assessment and get your results instantly.

    Test Yourself

    Practice and master your preparation for a specific topic or chapter. Check you scores at the end of the test.
    Advertisement

    Principle of Mathematical Induction

    Multiple Choice QuestionsLong Answer Type

    1.

    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

    761 Views
    Answer
    2.

    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

    596 Views
    Answer
    3.

    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

    398 Views
    Answer
    4.

    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>

    376 Views
    Answer
    Advertisement
    Advertisement
    Zigya App

    5.

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

    fraction numerator 1 over denominator 3.5 end fraction space plus space fraction numerator 1 over denominator 5.7 end fraction space plus space fraction numerator 1 over denominator 7.9 space end fraction space plus space......... space plus space fraction numerator 1 over denominator left parenthesis 2 straight n plus 1 right parenthesis left parenthesis 2 straight n plus 3 right parenthesis end fraction space equals space fraction numerator straight n over denominator 3 space left parenthesis 2 straight n space plus space 3 right parenthesis end fraction

    Let P(n) : fraction numerator 1 over denominator 3.5 end fraction plus fraction numerator 1 over denominator 5.7 end fraction plus fraction numerator 1 over denominator 7.9 end fraction plus...... plus fraction numerator 1 over denominator left parenthesis 2 straight n plus 1 right parenthesis left parenthesis 2 straight n plus 3 right parenthesis end fraction space equals space fraction numerator straight n over denominator 3 space left parenthesis 2 straight n plus 3 right parenthesis end fraction

    I.   For n = 1,

         space space straight P left parenthesis 1 right parenthesis space colon space fraction numerator 1 over denominator 3.5 end fraction space equals space fraction numerator 1 over denominator 3 left parenthesis 2.1 space plus 3 right parenthesis end fraction space rightwards double arrow space space fraction numerator 1 over denominator 3.5 end fraction space equals space fraction numerator 1 over denominator 3.5 end fraction space rightwards double arrow space 1 over 15 space equals space 1 over 15

    ∴     P(1) is true

    II.  Let the statement be true for n = m,  straight m space element of space straight N

    ∴    P(m): fraction numerator 1 over denominator 3.5 end fraction space plus space fraction numerator 1 over denominator 5.7 end fraction plus space fraction numerator 1 over denominator 7.9 end fraction plus space.......... plus space fraction numerator 1 over denominator left parenthesis 2 straight m plus 1 right parenthesis left parenthesis 2 straight m plus 3 right parenthesis end fraction space equals space fraction numerator straight m over denominator 3 left parenthesis 2 straight m plus 3 right parenthesis end fraction                          ...(i)

    III.  For n = 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  fraction numerator 1 over denominator 3.5 end fraction plus fraction numerator 1 over denominator 5.7 end fraction plus fraction numerator 1 over denominator 7.9 end fraction plus......... plus fraction numerator 1 over denominator left parenthesis 2 straight m plus 3 right parenthesis space left parenthesis 2 straight m space plus space 5 right parenthesis end fraction space equals space fraction numerator straight m plus 1 over denominator 3 space left parenthesis 2 straight m space plus 5 right parenthesis end fraction

    or    fraction numerator 1 over denominator 3.5 end fraction plus fraction numerator 1 over denominator 5.7 end fraction plus fraction numerator 1 over denominator 7.9 end fraction plus....... plus fraction numerator 1 over denominator left parenthesis 2 straight m plus 1 right parenthesis space left parenthesis 2 straight m space plus space 3 right parenthesis end fraction space plus space fraction numerator 1 over denominator left parenthesis 2 straight m space plus space 3 right parenthesis space left parenthesis 2 straight m space plus space 5 right parenthesis end fraction space equals space fraction numerator straight m plus 1 over denominator 3 space left parenthesis 2 straight m space plus space 5 right parenthesis end fraction   ... (ii)

                                    (Note that the last but one term in (ii) is always the same as last term in (i)]

            From (i), fraction numerator 1 over denominator 3.5 end fraction plus fraction numerator 1 over denominator 5.7 end fraction plus fraction numerator 1 over denominator 7.9 end fraction plus....... plus fraction numerator 1 over denominator left parenthesis 2 straight m plus 1 right parenthesis space left parenthesis 2 straight m plus 3 right parenthesis end fraction space equals space fraction numerator straight m over denominator 3 space left parenthesis 2 straight m plus 3 right parenthesis end fraction

    ∴    straight P left parenthesis straight m plus 1 right parenthesis colon space fraction numerator straight m over denominator 3 space left parenthesis 2 straight m plus 3 right parenthesis end fraction plus space fraction numerator 1 over denominator left parenthesis 2 straight m plus 3 right parenthesis space left parenthesis 2 straight m space plus space 5 right parenthesis end fraction space equals space fraction numerator straight m plus 1 over denominator 3 space left parenthesis 2 straight m plus 5 right parenthesis end fraction

    rightwards double arrow space space space space space space space space fraction numerator 1 over denominator left parenthesis 2 straight m plus 3 right parenthesis end fraction open square brackets straight m over 3 plus fraction numerator 1 over denominator 2 straight m plus 5 end fraction close square brackets space equals space fraction numerator straight m plus 1 over denominator 3 space left parenthesis 2 straight m plus 5 right parenthesis end fraction space rightwards double arrow space fraction numerator 1 over denominator 2 straight m plus 3 end fraction open square brackets fraction numerator 2 straight m squared plus 5 straight m plus 3 over denominator 3 space left parenthesis 2 straight m space plus 5 right parenthesis end fraction close square brackets space equals space fraction numerator straight m plus 1 over denominator 3 left parenthesis 2 straight m space plus 5 right parenthesis end fraction

    space space rightwards double arrow space space space fraction numerator 1 over denominator 2 straight m plus 3 end fraction open square brackets fraction numerator left parenthesis 2 straight m plus 3 right parenthesis space left parenthesis straight m plus 1 right parenthesis over denominator 3 space left parenthesis 2 straight m space plus 5 right parenthesis end fraction close square brackets space equals space fraction numerator straight m plus 1 over denominator 3 space left parenthesis 2 straight m space plus space 5 right parenthesis end fraction rightwards double arrow space fraction numerator straight m plus 1 over denominator 3 space left parenthesis 2 straight m plus 5 right parenthesis end fraction space equals space fraction numerator straight m plus 1 over denominator 3 space left parenthesis 2 straight m plus 5 right parenthesis end fraction

            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 space straight N.

    329 Views
  • Switch Icon Switch
    • Advertisement
    6.

    Prove the following by 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>




    226 Views
    Answer
    7.

    Prove the following by using the principle of mathematical induction for allspace space space space 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>.

    221 Views
    Answer
    8.

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

    1.3 space plus space 2.3 squared space plus space 3.3 cubed space plus space........ space plus space straight n.3 to the power of straight n space equals space fraction numerator left parenthesis 2 straight n minus 1 right parenthesis space 3 to the power of straight n plus 1 end exponent space plus space 3 over denominator 4 end fraction


    189 Views
    Answer
    Advertisement
    9.

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

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

    173 Views
    Answer
    10.

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

    1 plus 3 plus 5 plus........... space plus space left parenthesis 2 straight n minus 1 right parenthesis space equals space straight n squared

    276 Views
    Answer
    Advertisement
    Flag Popup Icon
    Flag it
    Submit
    Flag Popup Icon
    Switch to
    flexipad
    Share It Icon
    Share it
    Share It
    Share it