document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1067828>Question 1067828</A>: <I><SPAN STYLE=\"word-break: break-all;\"> While on a vacation to the country, a man spent 10 days fishing in the river. On each new, he caught 4 fish more than he had caught the day before. If he caught a total of 200 fish how many did he catch on each day? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #683005 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=683005\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Do they want an average number of fish caught per day,<BR>\n" );
document.write( "do they want the number caught each day as a day-by-day list<BR>\n" );
document.write( "or do they want the number caught each day as a function or formula?<BR>\n" );
document.write( " <BR>\n" );
document.write( "This is a case of an arithmetic sequence (or arithmetic progression, as it is called in some places).<BR>\n" );
document.write( "Each new day the man catches <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" >  fish more than he had caught the day before<BR>\n" );
document.write( "That means that the number of fish he catches forms an arithmetic sequence (or arithmetic progression).<BR>\n" );
document.write( "Some teachers like to see written formulas so you can \"show your work,\"<BR>\n" );
document.write( "but if that is not required, this is a mental math problem,<BR>\n" );
document.write( "and you can find the answer quickly an easily,<BR>\n" );
document.write( "which is great if you are taking a timed test.\r<BR>\n" );
document.write( "\n" );
document.write( "THE FORMULAS<BR>\n" );
document.write( "Here is the way I would write them. It is a popular way, but may not be the way your teacher prefers.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B1%5D ALIGN=MIDDLE  ALT=\"a%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >= first term of the arithmetic sequence/progression (number of fish caught on day <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > )<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d ALIGN=MIDDLE  ALT=\"d\"   DISABLED_style= \"max-width:90%; height:auto;\" >= common difference (in this case <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=4 ALIGN=MIDDLE  ALT=\"d=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > , because each new day the man catches <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" >  fish more than he had caught the day before).<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bn%5D=a%5B1%5D%2Bd%28n-1%29 ALIGN=MIDDLE  ALT=\"a%5Bn%5D=a%5B1%5D%2Bd%28n-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = term number <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5Bn%5D=n%28a%5B1%5D%2Ba%5Bn%5D%29%2F2 ALIGN=MIDDLE  ALT=\"S%5Bn%5D=n%28a%5B1%5D%2Ba%5Bn%5D%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5Bn%5D=n%282a%5B1%5D%2Bd%28n-1%29%29%2F2 ALIGN=MIDDLE  ALT=\"S%5Bn%5D=n%282a%5B1%5D%2Bd%28n-1%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >= sum of the first <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > terms<BR>\n" );
document.write( " <BR>\n" );
document.write( "THE MENTAL MATH:<BR>\n" );
document.write( "My telepathic powers are down today,<BR>\n" );
document.write( "so I will have to write the mental procedure as equations/formulas.<BR>\n" );
document.write( "The nice thing about arithmetic sequences,<BR>\n" );
document.write( "is that the average of any number of consecutive terms of such a sequence<BR>\n" );
document.write( "is the average of the first and last terms, so<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bavg%5D=%28a%5B1%5D%2Ba%5B10%5D%29%2F2 ALIGN=MIDDLE  ALT=\"a%5Bavg%5D=%28a%5B1%5D%2Ba%5B10%5D%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >= average number of fish caught per day.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10%2Aa%5Bavg%5D=200 ALIGN=MIDDLE  ALT=\"10%2Aa%5Bavg%5D=200\"   DISABLED_style= \"max-width:90%; height:auto;\" >= total number of fish caught in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10 ALIGN=MIDDLE  ALT=\"10\"   DISABLED_style= \"max-width:90%; height:auto;\" > days.<BR>\n" );
document.write( "So, we see that the man caught an average of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bavg%5D=200%2F10=20 ALIGN=MIDDLE  ALT=\"a%5Bavg%5D=200%2F10=20\"   DISABLED_style= \"max-width:90%; height:auto;\" > fish per day.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28average_catch=20%29 ALIGN=MIDDLE  ALT=\"highlight%28average_catch=20%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=fish ALIGN=MIDDLE  ALT=\"fish\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=per ALIGN=MIDDLE  ALT=\"per\"   DISABLED_style= \"max-width:90%; height:auto;\" ><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=day ALIGN=MIDDLE  ALT=\"day\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B10%5D=a%5B1%5D%2B3%2810-1%29=a%5B1%5D%2B3%2A9=a%5B1%5D%2B36 ALIGN=MIDDLE  ALT=\"a%5B10%5D=a%5B1%5D%2B3%2810-1%29=a%5B1%5D%2B3%2A9=a%5B1%5D%2B36\"   DISABLED_style= \"max-width:90%; height:auto;\" >= number of fish caught on day <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10 ALIGN=MIDDLE  ALT=\"10\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "so<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bavg%5D=%28a%5B1%5D%2Ba%5B10%5D%29%2F2=%28a%5B1%5D%2Ba%5B1%5D%2B36%29%2F2=%282a%5B1%5D%2B36%29%2F2=a%5B1%5D%2B18 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We also see that he caught <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > fish the first day:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B1%5D%2B18=20 ALIGN=MIDDLE  ALT=\"a%5B1%5D%2B18=20\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B1%5D=20-18 ALIGN=MIDDLE  ALT=\"a%5B1%5D=20-18\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B1%5D=2 ALIGN=MIDDLE  ALT=\"a%5B1%5D=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "So, the list of fish caught per day is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28matrix%281%2C19%2C2%2C%22%2C%22%2C6%2C%22%2C%22%2C10%2C%22%2C%22%2C14%2C%22%2C%22%2C18%2C%22%2C%22%2C22%2C%22%2C%22%2C26%2C%22%2C%22%2C30%2C%22%2C%22%2C34%2C%22%2C%22%2C38%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The number of fish expressed as a function would be<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28a%5Bn%5D=2%2B4%28n-1%29%29 ALIGN=MIDDLE  ALT=\"highlight%28a%5Bn%5D=2%2B4%28n-1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "or using an equivalent expression,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28a%5Bn%5D=4n-2%29 ALIGN=MIDDLE  ALT=\"highlight%28a%5Bn%5D=4n-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > . </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );