document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1006085>Question 1006085</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Five hundred and twenty two circular cylinders are stacked in rows in<BR>\n" );
document.write( "such a way that the number in a row is one less than in the row below.<BR>\n" );
document.write( "The bottom row contains 32 cylinders. How many rows are there?<BR>\n" );
document.write( "How many cylinders are there in the top row? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #622383 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=622383\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> \r<BR>\n" );
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document.write( "you have two formulas to work with.\r<BR>\n" );
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document.write( "Sn = n * (A1 + An) / 2\r<BR>\n" );
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document.write( "An = A1 + (n-1) * d\r<BR>\n" );
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document.write( "Sn is the sum of the terms in the arithmetic series.<BR>\n" );
document.write( "An is the nth term of the arithmetic series.<BR>\n" );
document.write( "n is the number of terms in the arithmetic series.<BR>\n" );
document.write( "d is the commpon difference.<BR>\n" );
document.write( "A1 is the first term in the arithmetic series.\r<BR>\n" );
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document.write( "the first term in the series is the bottom row.\r<BR>\n" );
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document.write( "A1 = 32\r<BR>\n" );
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document.write( "the common difference is -1 because each succeeding row agove the first row has one less barrel in it.\r<BR>\n" );
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document.write( "d = -1\r<BR>\n" );
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document.write( "The sum of the terms in the arithmetic series is 522 barrels.\r<BR>\n" );
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document.write( "Sn = 522\r<BR>\n" );
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document.write( "Sn = n * (A1 + An) / 2 becomes:\r<BR>\n" );
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document.write( "522 = n * (A1 + An) / 2\r<BR>\n" );
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document.write( "multiply both sides of this formula by 2 to get:\r<BR>\n" );
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document.write( "1044 = n * (A1 + An)\r<BR>\n" );
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document.write( "replace A1 with 32 to get:\r<BR>\n" );
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document.write( "1044 = n * (32 + An) *****\r<BR>\n" );
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document.write( "hold on to this for now.\r<BR>\n" );
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document.write( "An = A1 + (n-1) * d becomes:\r<BR>\n" );
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document.write( "An = 32 + (n-1) * -1 which becomes:\r<BR>\n" );
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document.write( "An = 32 -n + 1 which becomes:\r<BR>\n" );
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document.write( "An = 33 - n *****\r<BR>\n" );
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document.write( "the two formulas you have to work with now are:\r<BR>\n" );
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document.write( "1044 = n * (32 + An)<BR>\n" );
document.write( "An = 33 - n\r<BR>\n" );
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document.write( "replace An in the first equation with 33 - n to get:\r<BR>\n" );
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document.write( "1044 = n * (32 + 33 - n)\r<BR>\n" );
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document.write( "distribute the multiplication to get:\r<BR>\n" );
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document.write( "1044 = 32n + 33n - n^2\r<BR>\n" );
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document.write( "combine like terms to get:\r<BR>\n" );
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document.write( "1044 = 65n - n^2\r<BR>\n" );
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document.write( "add n^2 to both sides of the equation and subtract 65n from both sides of the eqaution to get:\r<BR>\n" );
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document.write( "n^2 - 65n + 1044 = 0\r<BR>\n" );
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document.write( "factor this quadratic to get:\r<BR>\n" );
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document.write( "(n-36) * (n-29) = 0\r<BR>\n" );
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document.write( "solve for n to get:\r<BR>\n" );
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document.write( "n = 36<BR>\n" );
document.write( "n = 29\r<BR>\n" );
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document.write( "you have two possible solutions for n.\r<BR>\n" );
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document.write( "substitute in the An equation to get:\r<BR>\n" );
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document.write( "An = 33 - n\r<BR>\n" );
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document.write( "when n = 36, An = -3\r<BR>\n" );
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document.write( "since you can't have -3 barrels, this solution is no good.\r<BR>\n" );
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document.write( "when n = 29, An = 4\r<BR>\n" );
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document.write( "this solution looks good so we'll test it out in the Sn equation.\r<BR>\n" );
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document.write( "Sn = n * (A1 + An) / 2 becomes:\r<BR>\n" );
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document.write( "522 = 29 * (32 + 4) / 2 which becomes:\r<BR>\n" );
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document.write( "522 = (29 * (36) / 2 which becomes:\r<BR>\n" );
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document.write( "522 = 522 which confirms the solution of n = 29 and An = 4 is good.\r<BR>\n" );
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document.write( "  </SPAN>\n" );
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