Question 1173561
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The number of apples in layer k is<br>
1+2+3+...+(k-1)+k<br>
This sum is equal to<br>
(k(k+1))/2<br>
That expression is equal to<br>
C(k+1,2)<br>
So the sum we are looking for is<br>
C(2,2)+C(3,2)+C(4,2)+...+C(N,2)+C(N+1,2)<br>
Using the hockey stick identity for Pascal's Triangle, this sum is equal to<br>
{{{C(N+2,3) = ((N+2)(N+1)(N))/3!}}}<br>
Do an internet search if you aren't familiar with the hockey stick identity....<br>