document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=30794>Question 30794</A>: <I><SPAN STYLE=\"word-break: break-all;\"> This question is really confusing, my tutor ended up answering it in class but everyone was still kinda dazed when we left class.  Will you please try to explain or solve this in a manner that is within a students grasp.\r<BR>\n" );
document.write( "\n" );
document.write( "Question:<BR>\n" );
document.write( "When the first n positive integers are added together, their sum is given by:<BR>\n" );
document.write( "1/2n(n+1)\r<BR>\n" );
document.write( "\n" );
document.write( "a) Demonstrate that this result holds for the case  n=5.<BR>\n" );
document.write( "b) Find the value of n for which the sum is 105.<BR>\n" );
document.write( "c) What is the smallest value of n for which the sum exceeds 1000?\r<BR>\n" );
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document.write( "The chapter we are working on is based on factoring.  Please help. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #17535 by </B><A HREF=/tutors/aboutme.mpl?userid=josmiceli><B>josmiceli(19441)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=josmiceli><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=josmiceli><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=17535\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> a)Demonstrate that this result holds for the case  n=5.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29%2An%2A%28n%2B1%29+=+S ALIGN=MIDDLE  ALT=\"%281%2F2%29%2An%2A%28n%2B1%29+=+S\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29%2A5%2A6+=+S ALIGN=MIDDLE  ALT=\"%281%2F2%29%2A5%2A6+=+S\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=30%2F2=+S ALIGN=MIDDLE  ALT=\"30%2F2=+S\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S+=+15 ALIGN=MIDDLE  ALT=\"S+=+15\"   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+2+%2B+3+%2B+4+%2B+5+=+30 ALIGN=MIDDLE  ALT=\"1+%2B+2+%2B+3+%2B+4+%2B+5+=+30\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(can do in your head or with calculator)<BR>\n" );
document.write( "--------------------------<BR>\n" );
document.write( "b)Find the value of n for which the sum is 105.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29%2An%2A%28n%2B1%29+=+105 ALIGN=MIDDLE  ALT=\"%281%2F2%29%2An%2A%28n%2B1%29+=+105\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "multiply both sides by 2<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%2A%28n%2B1%29+=+210 ALIGN=MIDDLE  ALT=\"n%2A%28n%2B1%29+=+210\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "expand<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%5E2+%2B+n+=+210 ALIGN=MIDDLE  ALT=\"n%5E2+%2B+n+=+210\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%5E2+%2B+n+-210+=+0 ALIGN=MIDDLE  ALT=\"n%5E2+%2B+n+-210+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "use binomial formula<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+ ALIGN=MIDDLE  ALT=\"n+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "on <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Ax%5E2+%2B+b%2Ax+%2B+c+=+0 ALIGN=MIDDLE  ALT=\"a%2Ax%5E2+%2B+b%2Ax+%2B+c+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "a = 1<BR>\n" );
document.write( "b = 1<BR>\n" );
document.write( "c = -210<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+=+%28-1+%2B-+sqrt%28+1-4%2A%28-210%29%29%29%2F+2+ ALIGN=MIDDLE  ALT=\"n+=+%28-1+%2B-+sqrt%28+1-4%2A%28-210%29%29%29%2F+2+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+=+%28-1+%2B-+sqrt%28841%29%29%2F+2 ALIGN=MIDDLE  ALT=\"n+=+%28-1+%2B-+sqrt%28841%29%29%2F+2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+=+%28-1+%2B+29%29+%2F+2 ALIGN=MIDDLE  ALT=\"n+=+%28-1+%2B+29%29+%2F+2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+=+28%2F2 ALIGN=MIDDLE  ALT=\"n+=+28%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+=+14 ALIGN=MIDDLE  ALT=\"n+=+14\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(I checked this with calculator)<BR>\n" );
document.write( "--------------------<BR>\n" );
document.write( "c)<BR>\n" );
document.write( "What is the smallest value of n for which the sum exceeds 1000?<BR>\n" );
document.write( "let S = sum<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29%2An%2A%28n%2B1%29+=+S ALIGN=MIDDLE  ALT=\"%281%2F2%29%2An%2A%28n%2B1%29+=+S\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%5E2+%2B+n+=+2%2AS ALIGN=MIDDLE  ALT=\"n%5E2+%2B+n+=+2%2AS\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%5E2+%2B+n+-+2%2AS+=+0 ALIGN=MIDDLE  ALT=\"n%5E2+%2B+n+-+2%2AS+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "use binomial formula<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+ ALIGN=MIDDLE  ALT=\"n+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "on <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%2Ax%5E2+%2B+b%2Ax+%2B+c+=+0 ALIGN=MIDDLE  ALT=\"a%2Ax%5E2+%2B+b%2Ax+%2B+c+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "a = 1<BR>\n" );
document.write( "b = 1<BR>\n" );
document.write( "c = -2*S<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+=+%28-1+%2B-+sqrt%28+1%2B+4%2A2%2AS+%29%29%2F2 ALIGN=MIDDLE  ALT=\"n+=+%28-1+%2B-+sqrt%28+1%2B+4%2A2%2AS+%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2An+=+-1+%2B-+sqrt%28+1%2B+8%2AS+%29 ALIGN=MIDDLE  ALT=\"2%2An+=+-1+%2B-+sqrt%28+1%2B+8%2AS+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2An+%2B+1+=+sqrt%28+1%2B+8%2AS+%29 ALIGN=MIDDLE  ALT=\"2%2An+%2B+1+=+sqrt%28+1%2B+8%2AS+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "square both sides<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%2An+%2B+1%29%5E2+=+1+%2B+8%2AS ALIGN=MIDDLE  ALT=\"%282%2An+%2B+1%29%5E2+=+1+%2B+8%2AS\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "I want S to be a number just over 1000<BR>\n" );
document.write( "That would make <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%2An+%2B+1%29%5E2 ALIGN=MIDDLE  ALT=\"%282%2An+%2B+1%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > just over 8000<BR>\n" );
document.write( "that means <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2An+%2B+1 ALIGN=MIDDLE  ALT=\"2%2An+%2B+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > would be just over <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%288000%29 ALIGN=MIDDLE  ALT=\"sqrt%288000%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > which is<BR>\n" );
document.write( "about 89.4<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+2%2An+%2B+1+=+89.4 ALIGN=MIDDLE  ALT=\"+2%2An+%2B+1+=+89.4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2An+=+88.4 ALIGN=MIDDLE  ALT=\"2%2An+=+88.4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+=+44 ALIGN=MIDDLE  ALT=\"n+=+44\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "trying n = 44, I get<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29%2A44%2A45+=+990 ALIGN=MIDDLE  ALT=\"%281%2F2%29%2A44%2A45+=+990\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "that's a little low<BR>\n" );
document.write( "try n = 45<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29%2A45%2A46+=+1035 ALIGN=MIDDLE  ALT=\"%281%2F2%29%2A45%2A46+=+1035\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "this looks right. There may be a better way to do it- I just came up with this\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "  </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );