Question 1161157
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As suggested by tutor @ikleyn, there is a lot of fun mathematics in the patterns involved in solving this problem.<br>
Counting from the top row, the number of oranges in the n-th layer is the n-th triangular number, which is {{{C(n+1,2)}}}<br>
Those numbers are found in a diagonal of Pascal's Triangle.<br>
The hockey stick identity in Pascal's Triangle (another internet search for you, if you are not familiar with it), tells us that<br>
{{{sum(C(n+1,2),1,n)}}} = {{{C(n+2,3)}}}<br>
So the number of oranges in the stack of 50 layers is {{{C(52,3) = 22100}}}<br>