document.write( "Question 1096591: Use mathematical induction to prove 6 is a factor of n^3 + 3n^2 + 2n. Please pls pls pls help me. Thankyou. \n" ); document.write( "

Algebra.Com's Answer #711014 by ikleyn(52794) $\"\"$ $\"About$

You can put this solution on YOUR website!
.
\n" ); document.write( "Use mathematical induction to prove 6 is a factor of n^3 + 3n^2 + 2n. Please pls pls pls help me. Thank you.
\n" ); document.write( "~~~~~~~~~~~~~~~~~~~~~~\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "1.  According to the Method of Mathematical induction, check the statement at n = 1:\r\n" );
document.write( "\r\n" );
document.write( "    1^3 + 3*1^2 + 2^1 = 1 + 3 + 2 = 6 \r\n" );
document.write( "\r\n" );
document.write( "    and the statement is TRUE.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "2.  According to the Method of Mathematical induction,  let us assume that the statement is true for n= k, i.e. let assume that \r\n" );
document.write( "\r\n" );
document.write( "      is a multiple of 6.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    Consider the polynomial expression at  n = k+1. You have\r\n" );
document.write( "\r\n" );
document.write( "     =  = regroup the terms =  + .     (1)\r\n" );
document.write( "\r\n" );
document.write( "    According to the induction assumption, the term    is a multiple of 6.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    The last term   =  = 3*(k+1)*(k+2)  is the thrice the product of two consecutive integer numbers.\r\n" );
document.write( "\r\n" );
document.write( "    So, this product is multiple of 2, and when multiplied by 3, is a multiple of 6.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "    Thus, the right side of (1) is a multiple of 6,  and the induction step is proved.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "3.  According to the principle of the Mathematical induction, the original statement is proved.\r\n" );
document.write( "

\r
\n" ); document.write( "\n" ); document.write( "QED.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );