Sets

Sets

Question

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

space space 1 plus 2 plus 3 plus........ plus straight n less than 1 over 8 left parenthesis 2 straight n plus 1 right parenthesis squared

Answer

Let P(n):  1 + 2 + 3 + .......... +  n less than 1 over 8 left parenthesis 2 straight n space plus space 1 right parenthesis squared

I.
      For n = 1,
        straight P left parenthesis 1 right parenthesis space colon space 1 space less than 1 over 8 left parenthesis 2.1 space plus space 1 right parenthesis squared rightwards double arrow space 1 space less than space 1 over 8 left parenthesis 2 plus 1 right parenthesis squared rightwards double arrow 1 less than 1 over 8 left parenthesis 3 right parenthesis squared rightwards double arrow space 1 less than space 9 over 8

∴      P (1) is true
II.    Suppose the statement is true for n = m,  straight m element of space straight N
rightwards double arrow   straight P left parenthesis straight m right parenthesis space colon space 1 space plus space 2 space plus space 3 space plus............ space plus space straight m space less than space 1 over 8 left parenthesis 2 straight m space plus space 1 right parenthesis squared                         ... (i)

III.   
For n = m + 1,
rightwards double arrow   straight P space left parenthesis straight m space plus space 1 right parenthesis colon space 1 space plus space 2 space plus space 3 space plus space....... space plus space left parenthesis straight m space plus space 1 right parenthesis space less than space 1 over 8 left square bracket 2 space left parenthesis straight m plus 1 right parenthesis space plus space 1 right square bracket squared      ... (ii)
or       1 plus 2 plus 3 space plus space.......... plus space straight m space plus space left parenthesis straight m plus 1 right parenthesis space less than space 1 over 8 left parenthesis 2 straight m plus 3 right parenthesis squared
          From (i),
          1 space plus space 2 space plus 3 space plus space....... space plus space straight m space less than space 1 over 8 left parenthesis 2 straight m space plus space 1 right parenthesis squared
Adding (m + 1) on both sides, we get
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#6 {main}</pre>
rightwards double arrow 1 + 2 + 3 +........+ m + (m + 1) less than space 1 over 8 left square bracket 4 m squared plus 4 m plus 1 plus 8 m plus 8 right square bracket
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#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>
rightwards double arrow 1 + 2 + 3 + ......... + (m + 1) less than space 1 over 8 left parenthesis 2 straight m space plus space 3 right parenthesis squared

∴   P (m + 1) is true.

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


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