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The students in a school are having a cup stacking contest. John has a pyramid 9 layers high. How many cups will he need to add to his pyramid to increase the number of layers to 12?

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Please see the document enclosed.

If the stacking is done as per that method, then we need, for a stack of L levels,

total number of cups at level L = L

total number of cups for all levels from 1 to L : 1 + 2 + 3 + ... + L

= L ( L + 1)/2

So the number of cups needed to increase the number of layers to 12 from 9 is:

12(12+1)/2 - 9 (9+1)/2 = 78 - 45 = 33

====================

Another method is:

We need L more cups to increase the number of layers from L - 1 to L. Hence, we need, to go from level 9 to level 12: 10+11+12 = 33 cups more.

If the stacking is done as per that method, then we need, for a stack of L levels,

total number of cups at level L = L

total number of cups for all levels from 1 to L : 1 + 2 + 3 + ... + L

= L ( L + 1)/2

So the number of cups needed to increase the number of layers to 12 from 9 is:

12(12+1)/2 - 9 (9+1)/2 = 78 - 45 = 33

====================

Another method is:

We need L more cups to increase the number of layers from L - 1 to L. Hence, we need, to go from level 9 to level 12: 10+11+12 = 33 cups more.