document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=994585>Question 994585</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1,7,13.....are in ap.can the sum of any terms be 175?justify </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #613637 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=613637\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> For an AP (arithmetic progression or arithmetic sequence) with first term <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;\" > and common difference <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d ALIGN=MIDDLE  ALT=\"d\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "the 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 is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%5B1%5D%2Ba%5Bn%5D%29%2An%2F2 ALIGN=MIDDLE  ALT=\"%28a%5B1%5D%2Ba%5Bn%5D%29%2An%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > or<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%282a%5B1%5D%2B%28n-1%29%2Ad%29%2F2=n%282a%5B1%5D%29%2F2%2Bn%28n-1%29%2Ad%29%2F2=n%2Aa%5B1%5D%2Bn%28n-1%29%2F2 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .\r<BR>\n" );
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document.write( "The 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 cannot be 175.<BR>\n" );
document.write( " <BR>\n" );
document.write( "ONE APPROACH:<BR>\n" );
document.write( "For the AP 1, 7, 13, ... , with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28a%5B1%5D=1%2C+d=7-1=6%29 ALIGN=MIDDLE  ALT=\"system%28a%5B1%5D=1%2C+d=7-1=6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "the 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 is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%282%2A1%2B%28n-1%29%2A6%29%2F2=n%282%2B6%28n-1%29%29%2F2=n%282%2B6n-6%29%29%2F2=n%286n-4%29%29%2F2=n%283n-2%29 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=n%283n-2%29=175 ALIGN=MIDDLE  ALT=\"n%283n-2%29=175\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3n%5E2-2n=175 ALIGN=MIDDLE  ALT=\"3n%5E2-2n=175\"   DISABLED_style= \"max-width:90%; height:auto;\" >--><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3n%5E2-2n-175=0 ALIGN=MIDDLE  ALT=\"3n%5E2-2n-175=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The solution to that equation is not a valid <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > , because it is not an integer:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=%28-%28-2%29+%2B-+sqrt%28%28-2%29%5E2-4%2A3%2A%28-175%29+%29%29%2F%282%2A3%29=%282+%2B-+sqrt%284%2B2100+%29%29%2F6=%282+%2B-+sqrt%282104%29%29%2F6+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( " <BR>\n" );
document.write( "ANOTHER APPROACH:<BR>\n" );
document.write( "The 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 would be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2Ba%5Bn%5D%29%2An%2F2=%28%281%2Ba%5Bn%5D%29%2F2%29%2An ALIGN=MIDDLE  ALT=\"%281%2Ba%5Bn%5D%29%2An%2F2=%28%281%2Ba%5Bn%5D%29%2F2%29%2An\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "That means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2Ba%5Bn%5D%29%2F2 ALIGN=MIDDLE  ALT=\"%281%2Ba%5Bn%5D%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > are integers that multiply to yield <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=175 ALIGN=MIDDLE  ALT=\"175\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "We know that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2Ba%5Bn%5D%29%2F2%3En ALIGN=MIDDLE  ALT=\"%281%2Ba%5Bn%5D%29%2F2%3En\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "because <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%5B1%5D%2Ba%5B2%5D%29%2F2=%281%2B7%29%2F2=8%2F2=4%3E2 ALIGN=MIDDLE  ALT=\"%28a%5B1%5D%2Ba%5B2%5D%29%2F2=%281%2B7%29%2F2=8%2F2=4%3E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > ,<BR>\n" );
document.write( "and for each unit increase in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > , <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2Ba%5Bn%5D%29%2F2 ALIGN=MIDDLE  ALT=\"%281%2Ba%5Bn%5D%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > increases by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%2F2=6%2F2=3 ALIGN=MIDDLE  ALT=\"d%2F2=6%2F2=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "That makes the possible combinations of factors of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=175 ALIGN=MIDDLE  ALT=\"175\"   DISABLED_style= \"max-width:90%; height:auto;\" > are<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2A35=175 ALIGN=MIDDLE  ALT=\"5%2A35=175\"   DISABLED_style= \"max-width:90%; height:auto;\" > , which would mean <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28n=5%2C1%2Ba%5Bn%5D%29%2F2=35%29 ALIGN=MIDDLE  ALT=\"system%28n=5%2C1%2Ba%5Bn%5D%29%2F2=35%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%2A25=175 ALIGN=MIDDLE  ALT=\"7%2A25=175\"   DISABLED_style= \"max-width:90%; height:auto;\" > , which would mean <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28n=7%2C1%2Ba%5Bn%5D%29%2F2=25%29 ALIGN=MIDDLE  ALT=\"system%28n=7%2C1%2Ba%5Bn%5D%29%2F2=25%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "Neither option works:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=5 ALIGN=MIDDLE  ALT=\"n=5\"   DISABLED_style= \"max-width:90%; height:auto;\" > means <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bn%5D=a%5B5%5D=1%2B4%2A6=1%2B24=25 ALIGN=MIDDLE  ALT=\"a%5Bn%5D=a%5B5%5D=1%2B4%2A6=1%2B24=25\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2Ba%5Bn%5D%29%2F2=%281%2Ba%5B5%5D%29%2F2=%281%2B25%29%2F2=13 ALIGN=MIDDLE  ALT=\"%281%2Ba%5Bn%5D%29%2F2=%281%2Ba%5B5%5D%29%2F2=%281%2B25%29%2F2=13\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=7 ALIGN=MIDDLE  ALT=\"n=7\"   DISABLED_style= \"max-width:90%; height:auto;\" > means <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bn%5D=a%5B7%5D=1%2B6%2A6=1%2B36=37 ALIGN=MIDDLE  ALT=\"a%5Bn%5D=a%5B7%5D=1%2B6%2A6=1%2B36=37\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2Ba%5Bn%5D%29%2F2=%281%2Ba%5B7%5D%29%2F2=%281%2B37%29%2F2=19 ALIGN=MIDDLE  ALT=\"%281%2Ba%5Bn%5D%29%2F2=%281%2Ba%5B7%5D%29%2F2=%281%2B37%29%2F2=19\"   DISABLED_style= \"max-width:90%; height:auto;\" > .\r<BR>\n" );
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document.write( "A DIFFERENT INTERPRETATION OF THE PROBLEM:<BR>\n" );
document.write( "The sum of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n ALIGN=MIDDLE  ALT=\"n\"   DISABLED_style= \"max-width:90%; height:auto;\" > consecutive terms (not starting from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B1%5D=1 ALIGN=MIDDLE  ALT=\"a%5B1%5D=1\"   DISABLED_style= \"max-width:90%; height:auto;\" > ) could be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=175 ALIGN=MIDDLE  ALT=\"175\"   DISABLED_style= \"max-width:90%; height:auto;\" > :<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7%2B13%2B19%2B25%2B31%2B37%2B43=175 ALIGN=MIDDLE  ALT=\"7%2B13%2B19%2B25%2B31%2B37%2B43=175\"   DISABLED_style= \"max-width:90%; height:auto;\" > . </SPAN>\n" );
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