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 fraction numerator 1 over denominator straight n plus 1 end fraction space plus space fraction numerator 1 over denominator straight n plus 2 end fraction space plus space.......... space plus space fraction numerator 1 over denominator 2 straight n end fraction greater than 13 over 24


Answer

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I.       For n = 2(note this step, n>1)
        straight P left parenthesis 2 right parenthesis colon space space fraction numerator 1 over denominator 2 plus 1 end fraction plus fraction numerator 1 over denominator 2 plus 2 end fraction greater than 13 over 24
space space rightwards double arrow space space space 1 third space plus space 1 fourth greater than 13 over 24 space space rightwards double arrow space 7 over 12 greater than 13 over 24 space space space rightwards double arrow space 14 over 24 greater than 13 over 24
        which is true

∴       P(n) is true for n = 2
II.      Suppose the statement is true for n = m, straight m space element of space straight N comma space space straight m greater than 1.
rightwards double arrow   space space straight P left parenthesis straight m right parenthesis space equals space fraction numerator 1 over denominator straight m plus 1 end fraction plus fraction numerator 1 over denominator straight m plus 2 end fraction plus....... plus fraction numerator 1 over denominator 2 straight m end fraction greater than 13 over 24                      .... (i)
III.      For n = m + 1,
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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)
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or      fraction numerator 1 over denominator straight m plus 2 end fraction space plus space fraction numerator 1 over denominator straight m plus 3 end fraction plus....... plus fraction numerator 1 over denominator 2 straight m plus 2 end fraction greater than 13 over 24
or       space space fraction numerator 1 over denominator straight m plus 2 end fraction plus fraction numerator 1 over denominator straight m plus 3 end fraction plus..... plus fraction numerator 1 over denominator 2 straight m end fraction plus fraction numerator 1 over denominator 2 straight m plus 1 end fraction plus fraction numerator 1 over denominator 2 straight m plus 2 end fraction greater than 13 over 24
         Adding fraction numerator 1 over denominator 2 straight m plus 1 end fraction plus fraction numerator 1 over denominator 2 straight m plus 2 end fraction on both sides of (i), we get
 
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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)
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rightwards double arrow     <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)
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rightwards double arrow      fraction numerator 1 over denominator straight m plus 2 end fraction plus fraction numerator 1 over denominator straight m plus 3 end fraction plus....... plus fraction numerator 1 over denominator 2 straight m plus 2 end fraction greater than 13 over 24 plus space fraction numerator 2 straight m plus 2 plus 2 straight m plus 1 minus 4 straight m minus 2 over denominator 2 space left parenthesis 2 straight m plus 1 right parenthesis space left parenthesis straight m plus 1 right parenthesis end fraction
rightwards double arrow      fraction numerator 1 over denominator straight m plus 2 end fraction plus fraction numerator 1 over denominator straight m plus 3 end fraction plus......... plus fraction numerator 1 over denominator 2 straight m plus 2 end fraction greater than 13 over 24 plus fraction numerator 1 over denominator 2 left parenthesis straight m plus 1 right parenthesis space left parenthesis 2 straight m plus 1 right parenthesis end fraction
         But, fraction numerator 1 over denominator 2 space left parenthesis straight m plus 1 right parenthesis space left parenthesis 2 straight m plus 1 right parenthesis end fraction greater than 0

∴      space space fraction numerator 1 over denominator straight m plus 2 end fraction space plus space fraction numerator 1 over denominator straight m plus 3 end fraction plus......... plus fraction numerator 1 over denominator 2 straight m plus 2 end fraction greater than 13 over 24

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


           

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