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One hundred fifty posts are stacked in layers.  Each layer has one less post than the layer below.  Top layer has three posts.  

Find number of layers.

Thank you.

This is a variation on the method Gauss used (when he was a child!) to add the integers from 1 to 100.

Suppose there are n layers of posts. You can verify inductively that the bottom layer has (n+2) posts.

Add up the posts from each layer:
3 + 4 + ... + (n+1) + (n+2) = 150.

List the sum in reverse order:
(n+2) + (n+1) + ... + 4 + 3 = 150

Add the equations together (this might be easier to read:
(n+5) + (n+5) + ... + (n+5) = 300
n(n+5) = 300
nē + 5n - 300 = 0

Use the quadratic formula to solve for n:
n = -20, 15
Since there can't be a negative number of layers, n = 15.

There are 15 layers of posts.

I checked this in a spreadsheet:  

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