Barrack  puts tomatoes in piles containing different numbers of tomatoes.  The first three piles contain 1, 15 and 70 tomatoes.  The numbers in the sequence are the sum of the first triangular number, the first, second and third triangular numbers and first, second, third, fourth and fifth triangular number and so on.  How many tomatoes will be in each of the fifth and seventh  piles?

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The triangular numbers are 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, ... each of which is the sum of  natural numbers: e.g., 15=5+4+3+2+1. The sums of the triangular numbers are, 1, 4, 10, 20, 35, 56, 84, ... Therefore the sum of the first 3 is 10, the sum of the first 5 is 35, the sum of the first 7 is 84, first 9 165, first 11 286. However, the numbers 15 and 70 don’t match these. 15 is the 5th triangular number, but 70 isn’t a triangular number, but is the sum of 15 and 55 (5th and 10th triangular numbers).

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