document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1104787>Question 1104787</A>: <I><SPAN STYLE=\"word-break: break-all;\"> use the principals of mathematical induction to prove the following statement 1*2+3*4+5*6...+(2n-1)*(2n)=(1/3)n(n+1)(4n-1) </SPAN>\n" );
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document.write( "The formula to be proved is<br><BR>\n" );
document.write( "1*2+3*4+5*6...+(2n-1)*(2n)=(1/3)n(n+1)(4n-1)<br><BR>\n" );
document.write( "Step 1: show the formula is true for n=1.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2A2+=+%281%2F3%29%281%29%282%29%283%29 ALIGN=MIDDLE  ALT=\"1%2A2+=+%281%2F3%29%281%29%282%29%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2+=+2 ALIGN=MIDDLE  ALT=\"2+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Yes, the formula is true for n=1.<br><BR>\n" );
document.write( "Step 2: Assume the formula is true for n=k and show that it follows that it is also true for n=k+1.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2A2%2B3%2A4%2B5%2A6 ALIGN=MIDDLE  ALT=\"1%2A2%2B3%2A4%2B5%2A6\"   DISABLED_style= \"max-width:90%; height:auto;\" > ... +<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282k-1%29%2A%282k%29+%2B+%282k%2B1%29%282k%2B2%29 ALIGN=MIDDLE  ALT=\"%282k-1%29%2A%282k%29+%2B+%282k%2B1%29%282k%2B2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > =<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%281%2F3%29k%28k%2B1%29%284k-1%29%29+%2B+%282k%2B1%29%282k%2B2%29 ALIGN=MIDDLE  ALT=\"%28%281%2F3%29k%28k%2B1%29%284k-1%29%29+%2B+%282k%2B1%29%282k%2B2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > =<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%281%2F3%29k%28k%2B1%29%284k-1%29%29+%2B+2%282k%2B1%29%28k%2B1%29 ALIGN=MIDDLE  ALT=\"%28%281%2F3%29k%28k%2B1%29%284k-1%29%29+%2B+2%282k%2B1%29%28k%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > =<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28k%2B1%29%28%281%2F3%29k%284k-1%29%2B4k%2B2%29 ALIGN=MIDDLE  ALT=\"%28k%2B1%29%28%281%2F3%29k%284k-1%29%2B4k%2B2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > =<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28k%2B1%29%28%281%2F3%29k%284k-1%29%2B%281%2F3%29%2812k%2B6%29%29 ALIGN=MIDDLE  ALT=\"%28k%2B1%29%28%281%2F3%29k%284k-1%29%2B%281%2F3%29%2812k%2B6%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > =<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28k%2B1%29%28%281%2F3%29%284k%5E2-k%2B12k%2B6%29%29 ALIGN=MIDDLE  ALT=\"%28k%2B1%29%28%281%2F3%29%284k%5E2-k%2B12k%2B6%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > =<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F3%29%28k%2B1%29%284k%5E2%2B11k%2B6%29 ALIGN=MIDDLE  ALT=\"%281%2F3%29%28k%2B1%29%284k%5E2%2B11k%2B6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > =<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F3%29%28k%2B1%29%28k%2B2%29%284k%2B3%29 ALIGN=MIDDLE  ALT=\"%281%2F3%29%28k%2B1%29%28k%2B2%29%284k%2B3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > =<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F3%29%28k%2B1%29%28%28k%2B1%29%2B1%29%284%28%28k%2B1%29-1%29%29 ALIGN=MIDDLE  ALT=\"%281%2F3%29%28k%2B1%29%28%28k%2B1%29%2B1%29%284%28%28k%2B1%29-1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "This last expression is the formula we were to prove, with \"k\" replaced by \"k+1\", so the proof my mathematical induction is complete.<br><BR>\n" );
document.write( "Note when doing a proof like this, it can be hard to see where to go with the algebraic manipulations.<br><BR>\n" );
document.write( "The key to seeing which direction to go can be seen in the third line of work above, where I factored \"(2k+2)\" into \"2(k+1)\".  I did this because I knew that, with a factor \"k\" in the formula for n=k, I would need a factor of \"(k+1)\" in the formula for n = k+1.<br><BR>\n" );
document.write( "Once I then factored out that common factor of (k+1), it was easy to see what I had to do with the rest of the expression. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );