document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1150514>Question 1150514</A>: <I><SPAN STYLE=\"word-break: break-all;\">  Clara organizes cans in triangular piles, where each row has one less can than the row below. For example, the pile of 15 cans has 5 cans in the bottom row and 4 cans in the row above it.\r<BR>\n" );
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document.write( "(a) There are 3240 cans in a pile. How many cans are in the bottom row?\r<BR>\n" );
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document.write( "(b)  There are S cans and they are organized in a triangular pile with 'n' cans in the bottom row. Show that n^2 +n - 2S= 0\r<BR>\n" );
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document.write( "(c) Clara has 2100 cans. Explain why she cant organize them in a triangular pile. </SPAN>\n" );
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document.write( "The number of cans in a pile with n cans on the bottom row is<br><BR>\n" );
document.write( "S = 1 + 2 + 3 + ... + (n-1) + n<br><BR>\n" );
document.write( "The sum of the integers from 1 to n is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=S+=+%28n%28n%2B1%29%29%2F2 ALIGN=MIDDLE  ALT=\"S+=+%28n%28n%2B1%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "(a) S = 3240<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3240+=+%28n%28n%2B1%29%29%2F2 ALIGN=MIDDLE  ALT=\"3240+=+%28n%28n%2B1%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6480+=+n%28n%2B1%29 ALIGN=MIDDLE  ALT=\"6480+=+n%28n%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Solve by inspection: 80(81) = 6480, so n = 80.<br><BR>\n" );
document.write( "(b) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n%5E2%2Bn-2S+=+n%5E2%2Bn-2%28%28n%28n%2B1%29%29%2F2%29+=+n%5E2%2Bn-%28n%5E2%2Bn%29+=+0 ALIGN=MIDDLE  ALT=\"n%5E2%2Bn-2S+=+n%5E2%2Bn-2%28%28n%28n%2B1%29%29%2F2%29+=+n%5E2%2Bn-%28n%5E2%2Bn%29+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "(c) S = 2100<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2100+=+%28n%28n%2B1%29%29%2F2 ALIGN=MIDDLE  ALT=\"2100+=+%28n%28n%2B1%29%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4200+=+n%28n%2B1%29 ALIGN=MIDDLE  ALT=\"4200+=+n%28n%2B1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The solution to that equation is not an integer, so there can't be that many cans in the pile.<br><BR>\n" );
document.write( " </SPAN>\n" );
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