document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=551085>Question 551085</A>: <I><SPAN STYLE=\"word-break: break-all;\"> for part (a) find the sum of the first n terms of the arithmetic series. for part (b) find n for the given sum Sn.\r<BR>\n" );
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document.write( "45. 3+8+13+18+23+...<BR>\n" );
document.write( "a. n=20<BR>\n" );
document.write( "b. Sn=366\r<BR>\n" );
document.write( "\n" );
document.write( "46. 50+42+34+26+18+...<BR>\n" );
document.write( "a. n=40<BR>\n" );
document.write( "b. Sn=182\r<BR>\n" );
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document.write( "47. -10+(-5)+0+5+10+...<BR>\n" );
document.write( "a. n=19<BR>\n" );
document.write( "b. Sn=375\r<BR>\n" );
document.write( "\n" );
document.write( "48. 34+31+28+25+22+...<BR>\n" );
document.write( "a. n=32<BR>\n" );
document.write( "b. Sn=-12\r<BR>\n" );
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document.write( "49. 2+9+16+23+30+...<BR>\n" );
document.write( "a. n=68<BR>\n" );
document.write( "b. Sn=1661\r<BR>\n" );
document.write( "\n" );
document.write( "50. 2+16+30+44+58+...<BR>\n" );
document.write( "a. n=24<BR>\n" );
document.write( "b. Sn=2178\r<BR>\n" );
document.write( "\n" );
document.write( "please show your work also so i know how to do it,<BR>\n" );
document.write( "thank you for ur help and happy new year! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #359427 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=359427\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> My best guess is that your textbook or class notes tell you that the nth term of an arithmetic sequence, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bn%5D ALIGN=MIDDLE  ALT=\"a%5Bn%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >, can be calculated based on the 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 the 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;\" >, as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Bn%5D=a%5B1%5D%2B%28n-1%29%2Ad ALIGN=MIDDLE  ALT=\"a%5Bn%5D=a%5B1%5D%2B%28n-1%29%2Ad\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "Then they must have told you that the sum of the first n terms can be calculated as<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5Bn%5D=n%2A%28a%5Bn%5D%2Ba%5B1%5D%29%2F2 ALIGN=MIDDLE  ALT=\"S%5Bn%5D=n%2A%28a%5Bn%5D%2Ba%5B1%5D%29%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=S%5Bn%5D=n%2Aa%5B1%5D%2Bn%28n-1%29%2Ad%2F2 ALIGN=MIDDLE  ALT=\"S%5Bn%5D=n%2Aa%5B1%5D%2Bn%28n-1%29%2Ad%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "SOLUTIONS FOR THE FIRST TWO PROBLEMS<BR>\n" );
document.write( "45. 3+8+13+18+23+...<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B1%5D=3 ALIGN=MIDDLE  ALT=\"a%5B1%5D=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the common difference is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=8-3=5 ALIGN=MIDDLE  ALT=\"d=8-3=5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "a. n=20 Since you are not given <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B20%5D ALIGN=MIDDLE  ALT=\"a%5B20%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >, use the second formula<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5B20%5D=20%2A3%2B20%2820-1%29%2A5%2F2=60%2B20%2A19%2A5%2F2=60%2B950=1010 ALIGN=MIDDLE  ALT=\"S%5B20%5D=20%2A3%2B20%2820-1%29%2A5%2F2=60%2B20%2A19%2A5%2F2=60%2B950=1010\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "b. Sn=366 I don't think I ever did this kind of problem. <BR>\n" );
document.write( "I suppose they could expect you to use a guess and check strategy, and you could try n=10, and figure out that the answer is between 10 and 20.<BR>\n" );
document.write( "Otherwise, it gets complicated, but here it goes:<BR>\n" );
document.write( "With <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5Bn%5D=n%2Aa%5B1%5D%2Bn%28n-1%29%2Ad%2F2 ALIGN=MIDDLE  ALT=\"S%5Bn%5D=n%2Aa%5B1%5D%2Bn%28n-1%29%2Ad%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > plus <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5Bn%5D=366 ALIGN=MIDDLE  ALT=\"S%5Bn%5D=366\"   DISABLED_style= \"max-width:90%; height:auto;\" > for this sequence, it means<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=366=n%2A3%2Bn%28n-1%29%2A5%2F2 ALIGN=MIDDLE  ALT=\"366=n%2A3%2Bn%28n-1%29%2A5%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and multiplying both sides times 2<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=732=6n%2Bn%28n-1%29%2A5 ALIGN=MIDDLE  ALT=\"732=6n%2Bn%28n-1%29%2A5\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=732=6n%2B%28n%5E2-n%29%2A5 ALIGN=MIDDLE  ALT=\"732=6n%2B%28n%5E2-n%29%2A5\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=732=6n%2B5n%5E2-5n ALIGN=MIDDLE  ALT=\"732=6n%2B5n%5E2-5n\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5n%5E2%2Bn-732=0 ALIGN=MIDDLE  ALT=\"5n%5E2%2Bn-732=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Applying the quadratic formula:<BR>\n" );
document.write( " <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+=+%28-1+%2B-+sqrt%28+1%5E2-4%2A5%2A%28-732%29+%29%29%2F%282%2A5%29=%28-1+%2B-+sqrt%28+1%2B14640%29%29%2F10=%28-1+%2B-+sqrt%2814641%29%29%2F10=%28-1+%2B-+121%29%2F10 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "That gives us <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=12 ALIGN=MIDDLE  ALT=\"n=12\"   DISABLED_style= \"max-width:90%; height:auto;\" >, our answer.<BR>\n" );
document.write( "We disregard the other solution because it's negative, and we were looking for a positive integer for <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( "46. 50+42+34+26+18+...<BR>\n" );
document.write( "has <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B1%5D=50 ALIGN=MIDDLE  ALT=\"a%5B1%5D=50\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=42-50=-8 ALIGN=MIDDLE  ALT=\"d=42-50=-8\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "That means <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B7%5D=a%5B1%5D%2B%287-1%29%2A%28-8%29=50-48=2 ALIGN=MIDDLE  ALT=\"a%5B7%5D=a%5B1%5D%2B%287-1%29%2A%28-8%29=50-48=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5B8%5D=a%5B7%5D-8=-6 ALIGN=MIDDLE  ALT=\"a%5B8%5D=a%5B7%5D-8=-6\"   DISABLED_style= \"max-width:90%; height:auto;\" >, with all the terms after that being negative.<BR>\n" );
document.write( "What's more the sum of the positive terms (the first 7 terms) is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5B7%5D=7%2A%28a%5B7%5D%2Ba%5B1%5D%29%2F2=7%2A%282%2B50%29%2F2=7%2A52%2F2=182 ALIGN=MIDDLE  ALT=\"S%5B7%5D=7%2A%28a%5B7%5D%2Ba%5B1%5D%29%2F2=7%2A%282%2B50%29%2F2=7%2A52%2F2=182\"   DISABLED_style= \"max-width:90%; height:auto;\" >, which answers part b by lucky guess. <BR>\n" );
document.write( "a. n=40<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5B40%5D=40%2A50%2B40%2840-1%29%2A%28-8%29%2F2=40%2A50%2B40%2A39%2A%28-8%29%2F2=2000-6240=-4240 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "b. Sn=182 We answered it by guess above, but if we need to solve quadratic equations again:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5Bn%5D=n%2Aa%5B1%5D%2Bn%28n-1%29%2Ad%2F2 ALIGN=MIDDLE  ALT=\"S%5Bn%5D=n%2Aa%5B1%5D%2Bn%28n-1%29%2Ad%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=182=n%2A50%2Bn%28n-1%29%2A%28-8%29%2F2 ALIGN=MIDDLE  ALT=\"182=n%2A50%2Bn%28n-1%29%2A%28-8%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=182=50n-4n%5E2%2B4n ALIGN=MIDDLE  ALT=\"182=50n-4n%5E2%2B4n\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=182=50n-4n%5E2%2B4n ALIGN=MIDDLE  ALT=\"182=50n-4n%5E2%2B4n\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=182=54n-4n%5E2 ALIGN=MIDDLE  ALT=\"182=54n-4n%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4n%5E2-54n%2B182=0 ALIGN=MIDDLE  ALT=\"4n%5E2-54n%2B182=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2n%5E2-27n%2B91=0 ALIGN=MIDDLE  ALT=\"2n%5E2-27n%2B91=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "If you are good at factoring you see that  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2n%5E2-27n%2B91=%282n-13%29%28n-7%29 ALIGN=MIDDLE  ALT=\"2n%5E2-27n%2B91=%282n-13%29%28n-7%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to find the solutions to the equation. Otherwise, you apply the quadratic formula. Either way, you find the answer to part b <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;\" >,<BR>\n" );
document.write( "and another solution to the equation, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=13%2F2 ALIGN=MIDDLE  ALT=\"13%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, that we discard because it's not a positive integer.<BR>\n" );
document.write( "FOR THE REST<BR>\n" );
document.write( "I hope you know how to do it now. I also hope that guess and check is an accepted and viable alternative to find the answer to part b.<BR>\n" );
document.write( "Here are the answers I calculated for problems 47-50<BR>\n" );
document.write( "47. -10+(-5)+0+5+10+...<BR>\n" );
document.write( "a. n=19 <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5B19%5D=665 ALIGN=MIDDLE  ALT=\"S%5B19%5D=665\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "b. Sn=375  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=15 ALIGN=MIDDLE  ALT=\"n=15\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5B15%5D=375 ALIGN=MIDDLE  ALT=\"S%5B15%5D=375\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "48. 34+31+28+25+22+...<BR>\n" );
document.write( "a. n=32 <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5B32%5D=-400 ALIGN=MIDDLE  ALT=\"S%5B32%5D=-400\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "b. Sn=-12 <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=24 ALIGN=MIDDLE  ALT=\"n=24\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5B24%5D=-12 ALIGN=MIDDLE  ALT=\"S%5B24%5D=-12\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "49. 2+9+16+23+30+...<BR>\n" );
document.write( "a. n=68 <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5B68%5D=16082 ALIGN=MIDDLE  ALT=\"S%5B68%5D=16082\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "b. Sn=1661 <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=22 ALIGN=MIDDLE  ALT=\"n=22\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5B22%5D=1661 ALIGN=MIDDLE  ALT=\"S%5B22%5D=1661\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "50. 2+16+30+44+58+...<BR>\n" );
document.write( "a. n=24 <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5B32%5D=3912 ALIGN=MIDDLE  ALT=\"S%5B32%5D=3912\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "b. Sn=2178 <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=18 ALIGN=MIDDLE  ALT=\"n=18\"   DISABLED_style= \"max-width:90%; height:auto;\" > <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S%5B18%5D=2178 ALIGN=MIDDLE  ALT=\"S%5B18%5D=2178\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );