document.write( "Question 909315: Prove by induction that 3^n ≥ 2n +1 for all positive integers. \n" ); document.write( "

Algebra.Com's Answer #551792 by josgarithmetic(39618) $\"\"$ $\"About$

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Prove $\"3%5En%3E=2n%2B1\"$ .\r
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\n" ); document.write( "\n" ); document.write( "n=1 case: $\"3%5E1%3E=2%2A1%2B1\"$
\n" ); document.write( " $\"3%3E=3\"$ , yes.\r
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\n" ); document.write( "\n" ); document.write( "ASSUME $\"highlight_green%283%5Ek%3E=2k%2B1%29\"$ to be true. k is any natural number.\r
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\n" ); document.write( "\n" ); document.write( "PROVE that k+1 case is true:
\n" ); document.write( " $\"3%5E%28k%2B1%29%3E=2%28k%2B1%29%2B1\"$
\n" ); document.write( " $\"3%5E%28k%2B1%29%3E=2k%2B2%2B1\"$
\n" ); document.write( " $\"3%5E%28k%2B1%29%3E=2k%2B3\"$
\n" ); document.write( " $\"3%2A3%5Ek%3E=2k%2B3\"$
\n" ); document.write( "Divide both sides by 3
\n" ); document.write( " $\"highlight_green%283%5Ek%3E=2k%2F3%2B1%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Compare this last found inequality with the ASSUMPTION statement inequality.
\n" ); document.write( " $\"%282%2F3%29k%3C2k\"$ as you can compare in the corresponding positions in each statement.
\n" ); document.write( "The latter highlighted-in-green statement is also then true if the assumption
\n" ); document.write( "statement is true.
\n" ); document.write( "PROVED.
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