document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1161157>Question 1161157</A>: <I><SPAN STYLE=\"word-break: break-all;\"> GREAT PYRAMID OF ORANGES <BR>\n" );
document.write( "A very bored grocer was stacking oranges one day.  She decided to stack them in a triangular pyramid.  There was one orange in the top layer, three oranges in the second layer, six oranges in the third layer, and so on.  Each layer except the top formed an equilateral triangle.  <BR>\n" );
document.write( "How many oranges would it take her to build a pyramid 50 layers high?<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 #784653 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=784653\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "As suggested by tutor @ikleyn, there is a lot of fun mathematics in the patterns involved in solving this problem.<br><BR>\n" );
document.write( "Counting from the top row, the number of oranges in the n-th layer is the n-th triangular number, which is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%28n%2B1%2C2%29 ALIGN=MIDDLE  ALT=\"C%28n%2B1%2C2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Those numbers are found in a diagonal of Pascal's Triangle.<br><BR>\n" );
document.write( "The hockey stick identity in Pascal's Triangle (another internet search for you, if you are not familiar with it), tells us that<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sum%28C%28n%2B1%2C2%29%2C1%2Cn%29 ALIGN=MIDDLE  ALT=\"sum%28C%28n%2B1%2C2%29%2C1%2Cn%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%28n%2B2%2C3%29 ALIGN=MIDDLE  ALT=\"C%28n%2B2%2C3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "So the number of oranges in the stack of 50 layers is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%2852%2C3%29+=+22100 ALIGN=MIDDLE  ALT=\"C%2852%2C3%29+=+22100\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
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