Sets

Sets

Question

Using principle of mathematical induction, show that straight n less than 2 to the power of straight n space space for space all space straight n space element of space straight N

Answer

Let P(n): space space straight n less than 2 to the power of straight n
I.     For n = 1, straight P left parenthesis 1 right parenthesis colon space 1 space less than space 2 to the power of 1 rightwards double arrow space 1 space less than space 2

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

∴     P(m) : straight m less than 2 to the power of straight m comma space space straight m space element of space straight N                                                 ...(i)
III.   For n = m + 1,
       straight P left parenthesis straight m space plus space 1 right parenthesis colon space straight m space plus space 1 space less than 2 to the power of straight m plus 1 end exponent                                             ...(ii)
       From (i), space space straight m less than 2 to the power of straight m

rightwards double arrow  space space 2. straight m less than 2 to the power of straight m.2 space rightwards double arrow space 2 straight m less than 2 to the power of straight m plus 1 end exponent rightwards double arrow space straight m plus straight m less than 2 to the power of straight m plus 1 end exponent                 ...(iii)
       But 1 less or equal than straight m comma space space straight m space element of space straight N                                                      ... (iv)
       Adding (iii) and (iv), we get
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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)
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∴     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 space straight N.

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