document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=741827>Question 741827</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Is there a formula to solve this question....The population of a community was 82,000 at the beginning of 2000.  Assuming a rate of growth of 1.6% per year since 2000, what will be the population be at the beginning of 2025?<BR>\n" );
document.write( "I can manually get the answer but would rather use a formula.  Is there one that will work?  I can't seem to get the formulas that I know to work. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #452167 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=452167\">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=t ALIGN=MIDDLE  ALT=\"t\"   DISABLED_style= \"max-width:90%; height:auto;\" >= years since the beginning of 2000<BR>\n" );
document.write( "At the beginning of 2000, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=0 ALIGN=MIDDLE  ALT=\"t=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the population is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%280%29=P%5B0%5D=82000 ALIGN=MIDDLE  ALT=\"P%280%29=P%5B0%5D=82000\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "At the beginning of 2001, 1 year has passed and the population has increased by 1.6%,<BR>\n" );
document.write( "meaning that the increase is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1.6%2F100=0.016 ALIGN=MIDDLE  ALT=\"1.6%2F100=0.016\"   DISABLED_style= \"max-width:90%; height:auto;\" > of the initial amount,<BR>\n" );
document.write( "so it has increased by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.016%2AP%5B0%5D=0.016%2A82000 ALIGN=MIDDLE  ALT=\"0.016%2AP%5B0%5D=0.016%2A82000\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "That means that at the beginning of 2001, when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=1 ALIGN=MIDDLE  ALT=\"t=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the population is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%5B0%5D%2B0.016P%5B0%5D=P%5B0%5D%2A%281%2B0.016%29=P%5B0%5D%2A1.016 ALIGN=MIDDLE  ALT=\"P%5B0%5D%2B0.016P%5B0%5D=P%5B0%5D%2A%281%2B0.016%29=P%5B0%5D%2A1.016\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "With numbers it would be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=82000%2A1.016=83312 ALIGN=MIDDLE  ALT=\"82000%2A1.016=83312\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "That number is not important. What's important is that the population number got multiplied times <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1.016=1%2B0.016 ALIGN=MIDDLE  ALT=\"1.016=1%2B0.016\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "With the passage of each year the population gets increased (multiplied) by a factor of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1.016 ALIGN=MIDDLE  ALT=\"1.016\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "After 2 years, it would have been multiplied by 1.016 again, and it would be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28P%5B0%5D%2A1.016%29%2A1.016=P%5B0%5D%2A1.016%5E2 ALIGN=MIDDLE  ALT=\"%28P%5B0%5D%2A1.016%29%2A1.016=P%5B0%5D%2A1.016%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "After 3 years, it would have been multiplied by 1.016 again, and it would be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28P%5B0%5D%2A1.016%5E2%29%2A1.016=P%5B0%5D%2A1.016%5E3 ALIGN=MIDDLE  ALT=\"%28P%5B0%5D%2A1.016%5E2%29%2A1.016=P%5B0%5D%2A1.016%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and so on.<BR>\n" );
document.write( "If you are studying sequences and series, your teacher would say that it is a geometric sequence with a common ratio of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1.016 ALIGN=MIDDLE  ALT=\"1.016\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "After <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t ALIGN=MIDDLE  ALT=\"t\"   DISABLED_style= \"max-width:90%; height:auto;\" > years the population will be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28t%29=P%5B0%5D%2A1.016%5Et ALIGN=MIDDLE  ALT=\"P%28t%29=P%5B0%5D%2A1.016%5Et\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "Using your initial population of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=82000 ALIGN=MIDDLE  ALT=\"82000\"   DISABLED_style= \"max-width:90%; height:auto;\" > at the beginning of 2000, it is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28P%28t%29=82000%2A1.016%5Et%29 ALIGN=MIDDLE  ALT=\"highlight%28P%28t%29=82000%2A1.016%5Et%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "At the beginning of 2025, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=t=25 ALIGN=MIDDLE  ALT=\"t=25\"   DISABLED_style= \"max-width:90%; height:auto;\" > and my calculator says that<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%2825%29=82000%2A1.016%5E25=highlight%28121943%29 ALIGN=MIDDLE  ALT=\"P%2825%29=82000%2A1.016%5E25=highlight%28121943%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (rounded)<BR>\n" );
document.write( " <BR>\n" );
document.write( "NOTE: That kind of growth is called exponential growth.<BR>\n" );
document.write( "If you use the growth factor per unit of time in the formula, the time will be the exponent.<BR>\n" );
document.write( "In mathematics they like to use powers of the irrational number <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e ALIGN=MIDDLE  ALT=\"e\"   DISABLED_style= \"max-width:90%; height:auto;\" > instead, so you see formulas like<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P%28t%29=P%5B0%5D%2Ae%5Ekt ALIGN=MIDDLE  ALT=\"P%28t%29=P%5B0%5D%2Ae%5Ekt\"   DISABLED_style= \"max-width:90%; height:auto;\" >, but if you do not know about <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e ALIGN=MIDDLE  ALT=\"e\"   DISABLED_style= \"max-width:90%; height:auto;\" > and natural logarithms, don't worry about it, because I do not think that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e ALIGN=MIDDLE  ALT=\"e\"   DISABLED_style= \"max-width:90%; height:auto;\" > would help your calculation at all.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );