document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=30099>Question 30099</A>: <I><SPAN STYLE=\"word-break: break-all;\"> an extra credit for my honors math class. i cant figure it out for the life of me...\r<BR>\n" );
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document.write( "derive the formula A=Pe^rt from the general formula for compounding interestL A=P(1+r/n)^n*t\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #16844 by </B><A HREF=/tutors/aboutme.mpl?userid=Fermat><B>Fermat(136)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Fermat><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=Fermat><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=16844\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+P%281%2Br%2Fn%29%5Ent ALIGN=MIDDLE  ALT=\"A+=+P%281%2Br%2Fn%29%5Ent\"   DISABLED_style= \"max-width:90%; height:auto;\" > - interest compounded periodically<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+Pe%5Ert ALIGN=MIDDLE  ALT=\"A+=+Pe%5Ert\"   DISABLED_style= \"max-width:90%; height:auto;\" > ------- interest compounded continuously<BR>\n" );
document.write( "When interest is compounded continously, this means letting the number of periods in a year increase without bound.<BR>\n" );
document.write( "In other words, we find the limit, as n tends to infinity of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+P%281%2Br%2Fn%29%5Ent ALIGN=MIDDLE  ALT=\"A+=+P%281%2Br%2Fn%29%5Ent\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Lim+%281%2Br%2Fn%29%5Ent ALIGN=MIDDLE  ALT=\"Lim+%281%2Br%2Fn%29%5Ent\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+-%3E+infty ALIGN=MIDDLE  ALT=\"n+-%3E+infty\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Binomial theorem<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2Bx%29%5En+=+1+%2B+nx+%2B+n%28n-1%29x%5E2%2F2+%2B+n%28n-1%29%28n-2%29x%5E3%2F3%21+%2B+n%28n-1%29%28n-2%29%28n-3%29x%5E4%2F4%21+... ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Applying the Binomial theorem to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2Br%2Fn%29%5Ent ALIGN=MIDDLE  ALT=\"%281%2Br%2Fn%29%5Ent\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1+%2B+%28nt%29%28r%2Fn%29+%2B+%28nt%29%28nt-1%29%28r%2Fn%29%5E2%2F2+%2B+%28nt%29%28nt-1%29%28nt-2%29%28r%2Fn%29%5E3%2F3%21+%2B+%28nt%29%28nt-1%29%28nt-2%29%28nt-3%29%28r%2Fn%29%5E4%2F4%21+... ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1+%2B+%28tr%29+%2B+%28t%29%28t-1%2Fn%29r%5E2%2F2+%2B+%28t%29%28t-1%2Fn%29%28t-2%2Fn%29r%5E3%2F3%21+%2B+%28t%29%28t-1%2Fn%29%28t-2%2Fn%29%28t-3%2Fn%29r%5E4%2F4%21+... ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n-%3E+infty ALIGN=MIDDLE  ALT=\"n-%3E+infty\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fn+-%3E+0 ALIGN=MIDDLE  ALT=\"1%2Fn+-%3E+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, giving<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1+%2B+tr+%2B+t%5E2r%5E2%2F2+%2B+t%5E3r%5E3%2F3%21+%2B+t%5E4r%5E4%2F4%21+... ALIGN=MIDDLE  ALT=\"1+%2B+tr+%2B+t%5E2r%5E2%2F2+%2B+t%5E3r%5E3%2F3%21+%2B+t%5E4r%5E4%2F4%21+...\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1+%2B+%28rt%29+%2B+%28rt%29%5E2%2F2+%2B+%28rt%29%5E3%2F3%21+%2B+%28rt%29%5E4%2F4%21+... ALIGN=MIDDLE  ALT=\"1+%2B+%28rt%29+%2B+%28rt%29%5E2%2F2+%2B+%28rt%29%5E3%2F3%21+%2B+%28rt%29%5E4%2F4%21+...\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and the above is the Maclaurin series for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E%28rt%29 ALIGN=MIDDLE  ALT=\"e%5E%28rt%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "i.e. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2Br%2Fn%29%5Ent+-%3E+e%5E%28rt%29 ALIGN=MIDDLE  ALT=\"%281%2Br%2Fn%29%5Ent+-%3E+e%5E%28rt%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+-%3E+infty ALIGN=MIDDLE  ALT=\"n+-%3E+infty\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We wanted to find the limit, as n tends to infinity of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+P%281%2Br%2Fn%29%5Ent ALIGN=MIDDLE  ALT=\"A+=+P%281%2Br%2Fn%29%5Ent\"   DISABLED_style= \"max-width:90%; height:auto;\" > and we have the limit, as n tends to infinity, of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2Br%2Fn%29%5Ent ALIGN=MIDDLE  ALT=\"%281%2Br%2Fn%29%5Ent\"   DISABLED_style= \"max-width:90%; height:auto;\" > which is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E%28rt%29 ALIGN=MIDDLE  ALT=\"e%5E%28rt%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. We substitute <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E%28rt%29 ALIGN=MIDDLE  ALT=\"e%5E%28rt%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+P%281%2Br%2Fn%29%5Ent ALIGN=MIDDLE  ALT=\"A+=+P%281%2Br%2Fn%29%5Ent\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "So for continuous compounding we can write,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+Pe%5E%28rt%29 ALIGN=MIDDLE  ALT=\"A+=+Pe%5E%28rt%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "========== </SPAN>\n" );
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