document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1173561>Question 1173561</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Suppose apples are in the shape of a pyramid. On the top layer, there is 1 apple. On the second layer, there are 1+2 apples. On the third layer, there is 1+2+3 apples. On the fourth layer, there 1+2+3+4 apples. If there are N number of layers, how many apples are there in total? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #798822 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=798822\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "The number of apples in layer k is<br><BR>\n" );
document.write( "1+2+3+...+(k-1)+k<br><BR>\n" );
document.write( "This sum is equal to<br><BR>\n" );
document.write( "(k(k+1))/2<br><BR>\n" );
document.write( "That expression is equal to<br><BR>\n" );
document.write( "C(k+1,2)<br><BR>\n" );
document.write( "So the sum we are looking for is<br><BR>\n" );
document.write( "C(2,2)+C(3,2)+C(4,2)+...+C(N,2)+C(N+1,2)<br><BR>\n" );
document.write( "Using the hockey stick identity for Pascal's Triangle, this sum is equal to<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%28N%2B2%2C3%29+=+%28%28N%2B2%29%28N%2B1%29%28N%29%29%2F3%21 ALIGN=MIDDLE  ALT=\"C%28N%2B2%2C3%29+=+%28%28N%2B2%29%28N%2B1%29%28N%29%29%2F3%21\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Do an internet search if you aren't familiar with the hockey stick identity....<br><BR>\n" );
document.write( " </SPAN>\n" );
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