document.write( "Question 1135570: Use mathematical induction to prove the statement is true for all positive integers n.\r
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Algebra.Com's Answer #753198 by ikleyn(52781) $\"\"$ $\"About$

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document.write( "Let’s prove it using principle of mathematical induction (PMI).\r\n" );
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document.write( "    P(n)=n^3+2n.\r\n" );
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document.write( "For n=1,\r\n" );
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document.write( "    P(1)=1+2=3 which is divisible by 3.\r\n" );
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document.write( "so the base of induction is established.\r\n" );
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document.write( "Now for n=k, assume that\r\n" );
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document.write( "    P(k)=k^3+2k\r\n" );
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document.write( "is divisible by 3.\r\n" );
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document.write( "Then for n=k+1,\r\n" );
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document.write( "    P(k+1)=(k+1)^3+2(k+1) = k^3+2k+3k^2+3k+3=P(k)+3(k^2+k+1)\r\n" );
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document.write( "Since we assumed P(k) to be divisible by 3, therefore P(k+1) is also divisible by 3.\r\n" );
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document.write( "Hence by PMI,  n^3+2n  is divisible by  3  for any integer positive n.\r\n" );
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