document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=965979>Question 965979</A>: <I><SPAN STYLE=\"word-break: break-all;\"> prove that the product of any three consecutive numbers is even. <BR>\n" );
document.write( "So far I have tried ,      (n)(n+1)(n+2)=n^3+3n^2+2n<BR>\n" );
document.write( "but there is no multiple of an even number. I'm not sure what kind of proof method it is but this would be a proof with the same method. i.e prove that the sum of three consecutive odd numbers is divisible by 3 ,       2n+1+2n+3+2n+5= 6n+9  3(2n+3)     is a multiple of three therefore it is divisible by three.\r<BR>\n" );
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document.write( "Any help would be appreciated, hope it is not too confusing <BR>\n" );
document.write( "Thanks<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #590410 by </B><A HREF=/tutors/aboutme.mpl?userid=tenkun><B>tenkun(12)</B></A><img src=\"/images/stars/stars-06.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=tenkun><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=590410\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> let the consecutive numbers are n,(n+1),(n+2)<BR>\n" );
document.write( "product = n*(n+1)*(n+2)\r<BR>\n" );
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document.write( "if n=1<BR>\n" );
document.write( "P(1) = 1*2*3=6 --Even<BR>\n" );
document.write( "Assume that P(n)=even then we have to prove that P(n+1)=(n+1)*(n+2)*(n+3) is also even\r<BR>\n" );
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document.write( "P(n+1)= (n+1)*(n+2)*(n+3)<BR>\n" );
document.write( "      =n*(n+1)*(n+2)+ 3*(n+1)*(n+2)<BR>\n" );
document.write( "we know that n*(n+1)*(n+2)is even ( assumed )<BR>\n" );
document.write( "we have to show that 3*(n+1)*(n+2) is also even <BR>\n" );
document.write( "if n is odd then n+1 is even<BR>\n" );
document.write( "either way n^2+3*n+2 is even<BR>\n" );
document.write( "so 3*(n+1)*(n+2)=3*(n^2+3*n+2) is also even<BR>\n" );
document.write( "we have shown that P(1) is even and that P(n+1)is even if P(n)is even.<BR>\n" );
document.write( "hence proved. </SPAN>\n" );
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