document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=893401>Question 893401</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In an arithmetic sequence, the sum of the first three terms is 18 and the sum of the squares of these terms is 126. Find the terms<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #541265 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=541265\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> let x = the number<BR>\n" );
document.write( "let d = the difference.\r<BR>\n" );
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document.write( "the formula for the arithmetic sequence is An = A1 + (n-1) * d\r<BR>\n" );
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document.write( "let x = A1<BR>\n" );
document.write( "let d = d<BR>\n" );
document.write( "let n start at 1 for the first member in the sequence.\r<BR>\n" );
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document.write( "your arithmetic sequence formula becomes:\r<BR>\n" );
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document.write( "A1 = x + (1-1) * d = x<BR>\n" );
document.write( "A2 = x + (2-1) * d = x + d<BR>\n" );
document.write( "A3 = x + (3-1) * d = x + 2d\r<BR>\n" );
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document.write( "the first 3 numbers in your sequence are:\r<BR>\n" );
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document.write( "x<BR>\n" );
document.write( "x + d<BR>\n" );
document.write( "x + 2d\r<BR>\n" );
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document.write( "the sum of these first 3 numbers is 18.\r<BR>\n" );
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document.write( "you get:\r<BR>\n" );
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document.write( "3x + 3d = 18\r<BR>\n" );
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document.write( "when you square these numbers, you get:\r<BR>\n" );
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document.write( "x^2          = x^2\r\n" );
document.write( "(x + d)^2    = x^2 + 2dx + d^2\r\n" );
document.write( "(x + 2d)^2   = x^2 + 4dx + 4d^2\r\n" );
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document.write( "the sum of the squared numbers is equal to:\r<BR>\n" );
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document.write( "3x^2 + 6dx + 5d^2\r<BR>\n" );
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document.write( "this sum is equal to 126, so you get:\r<BR>\n" );
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document.write( "3x^2 + 6dx + 5d^2 = 126\r<BR>\n" );
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document.write( "subtract 126 from both sides of this equation and you get:\r<BR>\n" );
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document.write( "3x^2 + 6dx + 5d^2 - 126 = 0\r<BR>\n" );
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document.write( "that quadratic equation is now in standard form and can be factored.\r<BR>\n" );
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document.write( "you have 2 equations that need to be solved simultaneously.\r<BR>\n" );
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document.write( "they are:\r<BR>\n" );
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document.write( "3x + 3d = 18<BR>\n" );
document.write( "3x^2 + 6dx + 5d^2 - 126 = 0\r<BR>\n" );
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document.write( "solve for d in the first equation and then substitute that value for d in the second equation.\r<BR>\n" );
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document.write( "3x + 3d = 18 is your first equation.<BR>\n" );
document.write( "solve for d to get:<BR>\n" );
document.write( "d = 6 - x\r<BR>\n" );
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document.write( "3x^2 + 6dx + 5d^2 - 126 = 0 is your second equation<BR>\n" );
document.write( "replace d with (6-x) in that equation to get:<BR>\n" );
document.write( "3x^2 + 6*(6-x)*x + 5*((6-x)^2) - 126 = 0<BR>\n" );
document.write( "simplify that equation to get:<BR>\n" );
document.write( "3x^2 + 36x - 6x^2 + 5*(36 - 12x + x^2) - 126 = 0<BR>\n" );
document.write( "simplify that equation further by distributing the multiplication to get:<BR>\n" );
document.write( "3x^2 + 36x - 6x^2 + 180 - 60x + 5x^2 - 126 = 0<BR>\n" );
document.write( "simplify that equation further by combining like terms to get:<BR>\n" );
document.write( "2x^2 - 24x + 54 = 0<BR>\n" );
document.write( "divide both sides of this equation by 2 to get:<BR>\n" );
document.write( "x^2 - 12x + 27 = 0<BR>\n" );
document.write( "factor this equation to get:<BR>\n" );
document.write( "(x-3) * (x-9) = 0\r<BR>\n" );
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document.write( "your solutions are either x = 3 or x = 9.\r<BR>\n" );
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document.write( "use the first equation of d = 6 - x to get:\r<BR>\n" );
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document.write( "when x = 3, d = 3<BR>\n" );
document.write( "when x = 9, d = -3\r<BR>\n" );
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document.write( "those are your potential solutions.\r<BR>\n" );
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document.write( "you need to confirm them by replacing x and d in the original equations with them to see if the equations hold true.\r<BR>\n" );
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document.write( "when x = 3, d = 3\r<BR>\n" );
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document.write( "3x + 3d = 18 becomes 9 + 9 = 18 which is true.\r<BR>\n" );
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document.write( "when x = 9, d = -3 and you get 3x + 3d = 27 - 9 = 18 which is also true.\r<BR>\n" );
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document.write( "the sum of the arithmetic sequence is true.\r<BR>\n" );
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document.write( "let's look at the individual sequence and then square the numbers to see if the squared sequence is also true.\r<BR>\n" );
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document.write( "when x = 3 and d = 3, your sequence is:\r<BR>\n" );
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document.write( "3<BR>\n" );
document.write( "6<BR>\n" );
document.write( "9\r<BR>\n" );
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document.write( "when x = 9 and d = -3, your sequence is:\r<BR>\n" );
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document.write( "9<BR>\n" );
document.write( "6<BR>\n" );
document.write( "3\r<BR>\n" );
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document.write( "since they are the same numbers only in reverse order, the sum of their squares will be the same.\r<BR>\n" );
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document.write( "3^3 = 9<BR>\n" );
document.write( "6^2 = 36<BR>\n" );
document.write( "9^2 = 81\r<BR>\n" );
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document.write( "9 + 36 + 81 = 126\r<BR>\n" );
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document.write( "the sum of the first 3 number in the sequence is equal to 18.<BR>\n" );
document.write( "the sum of their squares is equal to 126.\r<BR>\n" );
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document.write( "looks like both solutions work.\r<BR>\n" );
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document.write( "your solutions are:\r<BR>\n" );
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document.write( "x = 3 and d = 3<BR>\n" );
document.write( "or:<BR>\n" );
document.write( "x = 9 and d = -3\r<BR>\n" );
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