SOLUTION: Cannonball Pyramids Cannonballs can be stacked to form a pyramid with a triangular base. Five of these pyramids are shown below. Find the number of cannonballs in the 8th pyrami

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Question 1159309: Cannonball Pyramids
Cannonballs can be stacked to form a pyramid with a triangular
base. Five of these pyramids are shown below. Find the number
of cannonballs in the 8th pyramid.
a1 equals 1
a2 equals 4
a3 equals 10
a4 equals 20
a5 equals 35
Found 2 solutions by greenestamps, Alan3354:
Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!


There is no picture of the pyramids.... But we don't need the picture.

Find the pattern for the given 5 numbers and then continue the pattern to find the 6th, 7th, and 8th numbers.

The numbers of cannonballs in the pyramids form a sequence of integers. Find the differences between successive terms of the sequence, then the differences between those differences, and so on, until the differences are constant.

Then you can continue that sequence of constant integers and work backwards to find later terms in the sequence.

Here is an array showing the terms of the original sequence and the first, second, and third differences:
    1   4  10  20  35
      3   6  10  15
        3   4   5
          1   1

The third differences are constant. To find three more terms of the sequence, add three more 1's to that row and work backwards up the array of numbers.
    1   4  10  20  35
      3   6  10  15
        3   4   5
          1   1   1   1   1

    1   4  10  20  35
      3   6  10  15
        3   4   5   6   7   8
          1   1   1   1   1

    1   4  10  20  35
      3   6  10  15  21  28  36
        3   4   5   6   7   8
          1   1   1   1   1

    1   4  10  20  35  56  64  100
      3   6  10  15  21  28  36
        3   4   5   6   7   8
          1   1   1   1   1


The 8th pyramid will have 100 cannonballs.
Answer by Alan3354(69443)   (Show Source): You can put this solution on YOUR website!
Cannonballs can be stacked to form a pyramid with a triangular
base. Five of these pyramids are shown below. Find the number
of cannonballs in the 8th pyramid.
a1 equals 1
a2 equals 4
a3 equals 10
a4 equals 20
a5 equals 35
=============
I haven't tried it, but I don't think you can stack the way the other tutor did.
I think the single cannonball will be on the top.
(JK)
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