document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1115661>Question 1115661</A>: <I><SPAN STYLE=\"word-break: break-all;\"> use mathematical induction to prove that the statement is true for all positive integers. <BR>\n" );
document.write( "5+23+53+...6n^2 - 1=n^2 (2n+3) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #734075 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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/cgi-bin/show-face.mpl?user=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=734075\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "To prove by mathematical induction:<BR>\n" );
document.write( "(1) show the statement is true for some beginning value (usually 1, but not always); and<BR>\n" );
document.write( "(2) show that, assuming it is true for some integer n, it follows that it is true for n+1 also<br><BR>\n" );
document.write( "The statement is true for n=1: 5 = (1^2)(2(1)+3) = 1*5 = 5<br><BR>\n" );
document.write( "We need to show that if<br><BR>\n" );
document.write( "5+23+53+...+(6n^2-1) = n^2(2n+3)<br><BR>\n" );
document.write( "is true, then it follows algebraically that<br><BR>\n" );
document.write( "5+23+53+...+(6n^2-1)+(6(n+1)^2-1) = (n+1)^2(2(n+1)+3)<br><BR>\n" );
document.write( "On the left, replace the sum up to the (6n^2-1) term with the expression n^2(2n+3) and simplify.  And simplify the expression on the right.<br><BR>\n" );
document.write( "The proof is complete if the two expressions are the same.<br><BR>\n" );
document.write( "On the left...<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%5E2%282n%2B3%29%2B%286%28n%2B1%29%5E2-1%29 ALIGN=MIDDLE  ALT=\"n%5E2%282n%2B3%29%2B%286%28n%2B1%29%5E2-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > =<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2n%5E3%2B3n%5E2%2B6%28n%5E2%2B2n%2B1%29-1 ALIGN=MIDDLE  ALT=\"2n%5E3%2B3n%5E2%2B6%28n%5E2%2B2n%2B1%29-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > =<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2n%5E3%2B3n%5E2%2B6n%5E2%2B12n%2B6-1 ALIGN=MIDDLE  ALT=\"2n%5E3%2B3n%5E2%2B6n%5E2%2B12n%2B6-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > =<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2n%5E3%2B9n%5E2%2B12n%2B5 ALIGN=MIDDLE  ALT=\"2n%5E3%2B9n%5E2%2B12n%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "On the right...<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28n%2B1%29%5E2%282%28n%2B1%29%2B3%29 ALIGN=MIDDLE  ALT=\"%28n%2B1%29%5E2%282%28n%2B1%29%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=%28n%5E2%2B2n%2B1%29%282n%2B5%29 ALIGN=MIDDLE  ALT=\"%28n%5E2%2B2n%2B1%29%282n%2B5%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > =<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2n%5E3%2B5n%5E2%2B4n%5E2%2B10n%2B2n%2B5 ALIGN=MIDDLE  ALT=\"2n%5E3%2B5n%5E2%2B4n%5E2%2B10n%2B2n%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" > =<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2n%5E3%2B9n%5E2%2B12n%2B5 ALIGN=MIDDLE  ALT=\"2n%5E3%2B9n%5E2%2B12n%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The proof by mathematical induction is complete.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );