Sets

Sets

Question

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

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#6 {main}</pre>




Answer

Let straight P left parenthesis straight n right parenthesis colon space fraction numerator 1 over denominator 1.2.3 end fraction plus space fraction numerator 1 over denominator 2.3.4 end fraction plus fraction numerator 1 over denominator 3.4.5 end fraction plus space...... plus fraction numerator 1 over denominator straight n left parenthesis straight n plus 1 right parenthesis left parenthesis straight n plus 2 right parenthesis end fraction space equals space fraction numerator straight n left parenthesis straight n plus 3 right parenthesis over denominator 4 left parenthesis straight n plus 1 right parenthesis left parenthesis straight n plus 2 right parenthesis end fraction
I.     For n = 1,
      straight P left parenthesis 1 right parenthesis colon space fraction numerator 1 over denominator 1.2.3 end fraction space equals space fraction numerator 1 left parenthesis 1 plus 3 right parenthesis over denominator 4 left parenthesis 1 plus 1 right parenthesis left parenthesis 1 plus 2 right parenthesis end fraction rightwards double arrow 1 over 6 space equals space fraction numerator 4 over denominator 4 left parenthesis 2 right parenthesis left parenthesis 3 right parenthesis end fraction rightwards double arrow space 1 over 6 space equals space 1 over 6

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

∴     straight P left parenthesis straight m right parenthesis colon space fraction numerator 1 over denominator 1.2.3 end fraction plus fraction numerator 1 over denominator 2.3.4 end fraction plus fraction numerator 1 over denominator 3.4.5 end fraction plus........ plus space fraction numerator 1 over denominator straight m left parenthesis straight m plus 1 right parenthesis left parenthesis straight m plus 2 right parenthesis end fraction space equals space fraction numerator straight m left parenthesis straight m plus 3 right parenthesis over denominator 4 left parenthesis straight m plus 1 right parenthesis left parenthesis straight m plus 2 right parenthesis end fraction space space space... left parenthesis i right parenthesis
III.  For n = m +1,
      straight P left parenthesis straight m plus 1 right parenthesis colon space fraction numerator 1 over denominator 1.2.3 end fraction plus fraction numerator 1 over denominator 2.3.4 end fraction plus fraction numerator 1 over denominator 3.4.5 end fraction plus....... plus fraction numerator 1 over denominator left parenthesis straight m plus 1 right parenthesis left parenthesis straight m plus 2 right parenthesis left parenthesis straight m plus 3 right parenthesis end fraction equals fraction numerator left parenthesis straight m plus 1 right parenthesis left parenthesis straight m plus 1 plus 3 right parenthesis over denominator 4 left parenthesis straight m plus 1 plus 1 right parenthesis left parenthesis straight m plus 1 plus 2 right parenthesis end fraction
or
   fraction numerator 1 over denominator 1.2.3 end fraction plus fraction numerator 1 over denominator 2.3.4 end fraction plus fraction numerator 1 over denominator 3.4.5 end fraction plus........ plus fraction numerator 1 over denominator straight m left parenthesis straight m plus 1 right parenthesis left parenthesis straight m plus 2 right parenthesis end fraction plus fraction numerator 1 over denominator left parenthesis straight m plus 1 right parenthesis left parenthesis straight m plus 2 right parenthesis left parenthesis straight m plus 3 right parenthesis end fraction

space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space
                                                                                              space space equals fraction numerator left parenthesis straight m plus 1 right parenthesis left parenthesis straight m plus 4 right parenthesis over denominator 4 left parenthesis straight m plus 2 right parenthesis left parenthesis straight m plus 3 right parenthesis end fraction

From (i),
fraction numerator 1 over denominator 1.2.3 end fraction space plus space fraction numerator 1 over denominator 2.3.4 end fraction plus fraction numerator 1 over denominator 3.4.5 end fraction plus space........ plus space fraction numerator 1 over denominator straight m left parenthesis straight m plus 1 right parenthesis left parenthesis straight m plus 2 right parenthesis end fraction equals space fraction numerator straight m left parenthesis straight m plus 3 right parenthesis over denominator 4 left parenthesis straight m plus 1 right parenthesis left parenthesis straight m plus 2 right parenthesis end fraction

∴   straight P left parenthesis straight m plus 1 right parenthesis colon fraction numerator straight m left parenthesis straight m plus 3 right parenthesis over denominator 4 left parenthesis straight m plus 1 right parenthesis space left parenthesis straight m plus 2 right parenthesis end fraction plus fraction numerator 1 over denominator left parenthesis straight m plus 1 right parenthesis space left parenthesis straight m plus 2 right parenthesis space left parenthesis straight m plus 3 right parenthesis end fraction equals fraction numerator left parenthesis straight m plus 1 right parenthesis space left parenthesis straight m plus 4 right parenthesis over denominator 4 left parenthesis straight m plus 2 right parenthesis space left parenthesis straight m plus 3 right parenthesis end fraction
rightwards double arrow space space fraction numerator 1 over denominator left parenthesis straight m plus 1 right parenthesis space left parenthesis straight m plus 2 right parenthesis end fraction open square brackets fraction numerator straight m left parenthesis straight m plus 3 right parenthesis over denominator 4 end fraction plus fraction numerator 1 over denominator straight m plus 3 end fraction close square brackets space equals space fraction numerator left parenthesis straight m plus 1 right parenthesis space left parenthesis straight m plus 4 right parenthesis over denominator 4 space left parenthesis straight m plus 2 right parenthesis space left parenthesis straight m plus 3 right parenthesis end fraction
rightwards double arrowfraction numerator 1 over denominator left parenthesis straight m plus 1 right parenthesis space left parenthesis straight m plus 2 right parenthesis end fraction open square brackets fraction numerator straight m cubed plus 6 straight m squared plus 9 straight m plus 4 over denominator 4 left parenthesis straight m plus 3 right parenthesis end fraction close square brackets equals fraction numerator left parenthesis straight m plus 1 right parenthesis space left parenthesis straight m plus 4 right parenthesis over denominator 4 left parenthesis straight m plus 2 right parenthesis space left parenthesis straight m plus 3 right parenthesis end fraction
rightwards double arrow space space space fraction numerator 1 over denominator left parenthesis straight m plus 1 right parenthesis space left parenthesis straight m plus 2 right parenthesis end fraction open square brackets fraction numerator left parenthesis straight m plus 1 right parenthesis space left parenthesis straight m plus 1 right parenthesis space left parenthesis straight m plus 4 right parenthesis over denominator 4 space left parenthesis straight m plus 3 right parenthesis end fraction close square brackets space equals space fraction numerator left parenthesis straight m plus 1 right parenthesis space left parenthesis straight m plus 4 right parenthesis over denominator 4 space left parenthesis straight m plus 2 right parenthesis space left parenthesis straight m plus 3 right parenthesis end fraction
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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)
#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)
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#6 {main}</pre> which is true

∴  P(m + 1) is true

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


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