document.write( "Question 1026132: Use mathematical induction to prove the statement is true for all positive integers n.\r
\n" ); document.write( "\n" ); document.write( "The integer n^3 + 2n is divisible by 3 for every positive integer n.
\n" ); document.write( "please show work so I can understand it:)\r
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Algebra.Com's Answer #641400 by ikleyn(52788) $\"\"$ $\"About$

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\n" ); document.write( "Use mathematical induction to prove the statement is true for all positive integers n.\r
\n" ); document.write( "\n" ); document.write( "The integer n^3 + 2n is divisible by 3 for every positive integer n.
\n" ); document.write( "please show work so I can understand it:)
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document.write( "1. Let us check the statement for n = 1.\r\n" );
document.write( "   At n = 1   =  = 1 + 2 = 3  is divisible by n.\r\n" );
document.write( "   So, the base of induction is valid.\r\n" );
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document.write( "2. Let us assume that the statement is true for some n = k, in other words, we assume that    is divisible by k. \r\n" );
document.write( "   We will prove that then it is true for the next integer  n=k+1.\r\n" );
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document.write( "   For  n = k+1  we have\r\n" );
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document.write( "    =  =  +  =  + .   (1)\r\n" );
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document.write( "   The first addend in the right side (1) is divisible by 3 due to the inductive hypothesis. \r\n" );
document.write( "   The second addend in the right side is divisible by 3, because it is multiple of 3. \r\n" );
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document.write( "   Hence, the right side of (1) is divisible by 3.\r\n" );
document.write( "   Therefore, the left side,  is divisible by 3, too.\r\n" );
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document.write( "Thus the base of induction is proven to be valid, and the induction step is proven to be valid, too.\r\n" );
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document.write( "Due to the principle of the Mathematical induction, the statement \r\n" );
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document.write( "   \"for any positive integer n the number  is divisible by 3\"\r\n" );
document.write( "is proved.\r\n" );
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document.write( "The proof is completed.\r\n" );
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