document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=257916>Question 257916</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In an arithmetic series the sum of the first 9 terms is 162 and the sum of the first 12 terms is 288. Determine the series. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #189785 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=189785\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> sum of the terms of an arithmetic series is given by the formula:\r<BR>\n" );
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document.write( "s[n] = n * (a[1] + a[n])/2\r<BR>\n" );
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document.write( "s[n] = sum of the n terms of the sequence.<BR>\n" );
document.write( "a[1] = first term in the sequence.<BR>\n" );
document.write( "a[n] = nth term in the sequence.<BR>\n" );
document.write( "n = number of terms in the sequence.\r<BR>\n" );
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document.write( "the formuula for the nth term in the sequence is:\r<BR>\n" );
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document.write( "a[n] = a[1] + (n-1) * d\r<BR>\n" );
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document.write( "d is the common difference.\r<BR>\n" );
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document.write( "you will use both these formulas to find the answer.\r<BR>\n" );
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document.write( "you are given:\r<BR>\n" );
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document.write( "sum of the first 9 terms in a sequence is 162<BR>\n" );
document.write( "sum of the first 12 terms in the same sequence is 288.\r<BR>\n" );
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document.write( "the formnula for the sum of the first n terms in a sequence is:\r<BR>\n" );
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document.write( "s[n] = n * (a[1] + a[n])/2\r<BR>\n" );
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document.write( "substitute for the first 9 terms to get:\r<BR>\n" );
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document.write( "162 = 9 * (a[1] + a[9])/2 (first equation)\r<BR>\n" );
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document.write( "substitute for the first 12 terms to get:\r<BR>\n" );
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document.write( "288 = 12 * (a[1] + a[12])/2 (second equation)\r<BR>\n" );
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document.write( "multiply both sides of the first equation by 2 and divide both sides of the first equation by 9 to get:\r<BR>\n" );
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document.write( "a[1] + a[9] = 36 (third equation)\r<BR>\n" );
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document.write( "multiply both sides of the second equation by 2 and divide both sides of the second equation by 12 to get:\r<BR>\n" );
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document.write( "a[1] + a[12] = 48 (fourth equation)\r<BR>\n" );
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document.write( "subtract third equation from fourth equation to get:\r<BR>\n" );
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document.write( "a[12] - a[9] = 12\r<BR>\n" );
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document.write( "since 12 - 9 = 3, there are 3 terms between a[9] and a[12].\r<BR>\n" );
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document.write( "12/3 = 4 which means the common difference between each term is 4.\r<BR>\n" );
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document.write( "the nth term in each sequence is given by the formula:\r<BR>\n" );
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document.write( "a[n] = a[1] + (n-1) * d\r<BR>\n" );
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document.write( "we now know that d = 4\r<BR>\n" );
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document.write( "we solve for a[9] to get:\r<BR>\n" );
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document.write( "a[9] = a[1] + 8*4 = a[1] + 32\r<BR>\n" );
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document.write( "we take our first equation and plug in our new found values.\r<BR>\n" );
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document.write( "first equation is:\r<BR>\n" );
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document.write( "162 = 9 * (a[1] + a[9])/2 (first equation)\r<BR>\n" );
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document.write( "substitute a[1] + 32 for a[9] to get:\r<BR>\n" );
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document.write( "162 = 9 * (a[1] + a[1] + 32)/2\r<BR>\n" );
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document.write( "combine like terms to get:\r<BR>\n" );
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document.write( "162 = 9 * (2* a[1] + 32)/2\r<BR>\n" );
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document.write( "multiply both sides of this equation by 2 and divide both sides of this equation by 9 to get:\r<BR>\n" );
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document.write( "36 = 2 * a[1] + 32\r<BR>\n" );
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document.write( "subtract 32 from both sides of this equation to get:\r<BR>\n" );
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document.write( "36-32 = 2 * a[1] which becomes:\r<BR>\n" );
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document.write( "4 = 2 * a[1].\r<BR>\n" );
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document.write( "divide both sides of this equation by 2 to get:\r<BR>\n" );
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document.write( "a[1] = 2.\r<BR>\n" );
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document.write( "we now know a[1] and we know d so we can construct the sequence.\r<BR>\n" );
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document.write( "formula for nth term in the sequence is:\r<BR>\n" );
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document.write( "a[n] = a[1] + (n-1) * d\r<BR>\n" );
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document.write( "since d = 4,...\r<BR>\n" );
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document.write( "the 9th term in the sequence is:\r<BR>\n" );
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document.write( "a[9] = 2 + 8*4 = 2 + 32 = 34\r<BR>\n" );
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document.write( "the 12th term in the sequence is:\r<BR>\n" );
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document.write( "a[12] = 2 + 11*4 = 2 + 44 = 46\r<BR>\n" );
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document.write( "the sequence is:\r<BR>\n" );
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document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">         \r\n" );
document.write( "term:         <font color = red>1</font>  2  3   4   5   6   7   8   <font color = red>9</font>   10  11  <font color = red>12</font>\r\n" );
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document.write( "sequence:     <font color = red>2</font>  6  10  14  18  22  26  30  <font color = red>34</font>  38  42  <font color = red>46</font>\r\n" );
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document.write( "2 is the 1st term.<BR>\n" );
document.write( "34 is the 9th term.<BR>\n" );
document.write( "46 is the 12th term.\r<BR>\n" );
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document.write( "common difference is 4.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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