document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1137605>Question 1137605</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Prove the following using mathematical induction. Be sure to show the anchor as well as<BR>\n" );
document.write( "the full induction steps. Please show all the parts to the<BR>\n" );
document.write( "proof.<BR>\n" );
document.write( "3+7+11+...+(4n-1)=n(2n+1) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #755484 by </B><A HREF=/tutors/aboutme.mpl?userid=math_helper><B>math_helper(2461)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_helper><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=math_helper><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=755484\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Base case:  n=1<BR>\n" );
document.write( "LHS: 3<BR>\n" );
document.write( "RHS: 1(2*1+1) = 1(3) = 3<BR>\n" );
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document.write( "Hypothesis: 3+7+11+...+(4k-1) = k(2k+1)<BR>\n" );
document.write( "<br>\r<BR>\n" );
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document.write( "Must show the above is true for n=k+1\r<BR>\n" );
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document.write( "3+7+11+...+(4k-1)+(4(k+1)-1) =<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+green%283%2B7%2B11%29+ ALIGN=MIDDLE  ALT=\"+green%283%2B7%2B11%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+green%28matrix%281%2C3%2C%22%2B%22%2C%22...%22%2C%22%2B%22%29%29 ALIGN=MIDDLE  ALT=\"+green%28matrix%281%2C3%2C%22%2B%22%2C%22...%22%2C%22%2B%22%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+green%28%284k-1%29%29+ ALIGN=MIDDLE  ALT=\"+green%28%284k-1%29%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > + (4(k+1)-1)  \r<BR>\n" );
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document.write( "By hypothesis, the green terms are  k(2k+1), so we write<BR>\n" );
document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+k%282k%2B1%29+%2B+%284%28k%2B1%29-1%29+ ALIGN=MIDDLE  ALT=\"+k%282k%2B1%29+%2B+%284%28k%2B1%29-1%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282k%5E2+%2B+k%29+%2B+%284k%2B3%29+ ALIGN=MIDDLE  ALT=\"%282k%5E2+%2B+k%29+%2B+%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=+2k%5E2+%2B+5k+%2B+3+ ALIGN=MIDDLE  ALT=\"+2k%5E2+%2B+5k+%2B+3+\"   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%282k%2B3%29+ ALIGN=MIDDLE  ALT=\"+%28k%2B1%29%282k%2B3%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(n=k+1)<BR>\n" );
document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+++n%282n%2B1%29+ ALIGN=MIDDLE  ALT=\"+++n%282n%2B1%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >   DONE<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );