document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1011827>Question 1011827</A>: <I><SPAN STYLE=\"word-break: break-all;\"> write 3 different series each with 5 terms and each with a sum of 100. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #627598 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=627598\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> i'll use an arithmetic series because there are formulas that should allow me to figure out what numbers could be used.\r<BR>\n" );
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document.write( "Sn = 1/2 * n * (A1 + An)\r<BR>\n" );
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document.write( "n = 5, therefore:\r<BR>\n" );
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document.write( "S5 = 1/2 * 5 * (A1 + A5)\r<BR>\n" );
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document.write( "multiply both sides by 2 to get:\r<BR>\n" );
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document.write( "2 * S5 = 5 * (A1 + A5)\r<BR>\n" );
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document.write( "divide everything by 5 to get:\r<BR>\n" );
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document.write( "2 * S5 / 5 = A1 + A5\r<BR>\n" );
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document.write( "subtract A1 from both sides of the equation to get:\r<BR>\n" );
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document.write( "2 * S5 / 5 - A1 = A5\r<BR>\n" );
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document.write( "let S5 equal 100 to get:\r<BR>\n" );
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document.write( "2 * 100 / 5 - A1 = A5\r<BR>\n" );
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document.write( "simplify to get:\r<BR>\n" );
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document.write( "40 - A1 = A5\r<BR>\n" );
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document.write( "since An = A1 + (n-1)*d, then:\r<BR>\n" );
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document.write( "40 - A1 = A1 + (n-1)*d\r<BR>\n" );
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document.write( "subtract A1 from both sides of this equation to get:\r<BR>\n" );
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document.write( "400 - 2A1 = (n-1)*d\r<BR>\n" );
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document.write( "distribute the multiplication to get:\r<BR>\n" );
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document.write( "40 - 2A1 = (n-1)*d\r<BR>\n" );
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document.write( "since n = 5, then:\r<BR>\n" );
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document.write( "40 - 2A1 = 4*d\r<BR>\n" );
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document.write( "solve for A1 to get:\r<BR>\n" );
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document.write( "A1 = 40 - 4*d)/2\r<BR>\n" );
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document.write( "simplify to get:\r<BR>\n" );
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document.write( "A1 = 20 - 2d\r<BR>\n" );
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document.write( "if this formula is good, then all we need to do is find a value for d that will allow us to subtract from 20 and get an integer (positive or negative should work).  that value will be A1.\r<BR>\n" );
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document.write( "let's see if this works.\r<BR>\n" );
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document.write( "if d = 1, then A1 = 20 - 2 = 18.<BR>\n" );
document.write( "A5 = A1 + 4d = 18 + 4 * 1 = 22<BR>\n" );
document.write( "S5 = 1/2 * 5 * (18 + 22) = 1/2 * 5 * 40 = 1/2 * 200 = 100<BR>\n" );
document.write( "it works !!!!!\r<BR>\n" );
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document.write( "if d = 2, then A1 = 20 - 4 = 16<BR>\n" );
document.write( "A5 = 16 + 4*2 = 16 + 8 = 24<BR>\n" );
document.write( "S5 = 1/2 * 5 * (16 + 24) = 1/2 * 5 * 40 = 1/2 * 200 = 100\r<BR>\n" );
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document.write( "id d = 4, then A1 = 20 - 8 = 12<BR>\n" );
document.write( "A5 = 12 + 4*4 = 12 + 16 = 28<BR>\n" );
document.write( "S5 = 1/2 * 5 * (12 + 28) = 1/2 * 5 * 40 = 1/2 * 200 = 100\r<BR>\n" );
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document.write( "it worked for all three !!!!\r<BR>\n" );
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document.write( "i used 3 different arithmetic series.<BR>\n" );
document.write( "it was a matter of applying the formulas that come with arithmetic series.<BR>\n" );
document.write( "An = A1 + (n-1)*d<BR>\n" );
document.write( "Sn = n/2 * (A1 + An)\r<BR>\n" );
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document.write( "just to see if it will work with a negative number:\r<BR>\n" );
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document.write( "start with A1 = 20 - 2d.<BR>\n" );
document.write( "let d = 100.<BR>\n" );
document.write( "2d = 200<BR>\n" );
document.write( "20 - 200 = -180<BR>\n" );
document.write( "A1 = -180<BR>\n" );
document.write( "d = 100<BR>\n" );
document.write( "A5 = -180 + 4*100 = -180 + 400 = 220.<BR>\n" );
document.write( "S5 = 5/2 * (-180 + 220) = 5/2 * 5*40/2 = 5*20 = 100\r<BR>\n" );
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document.write( "it works again !!!!!\r<BR>\n" );
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document.write( "the sequence is -180, -80, 20, 120, 220<BR>\n" );
document.write( "add them up and you get 100.\r<BR>\n" );
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document.write( "you can probably do the same this with geometric series, and if you made up your own series rules, you might even be able to develop a formula using that.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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