Quandaries
and Queries |
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if i make a block pyramid and it puts a new perimeter around it every time, for example the first layer will be 1 block across (area=1), the second layer will be 3 blocks across (area=9), the third layer will be 5 blocks across (area=15),etc. The normal block pyramid. I have figured out that in order to figure out the number of blocks needed for a certain level, the equation is (2x-1) Sincerely, Kyle |
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Hi Kyle, To find the total number of blocks needed to build your pyramid to the level x you need to sum the squares of the odd integers from 1 to 2x - 1. The key to solving this problem is to know how to find the sum of all the integers from 1 to n. This expression is
I am going to write the sum on the left with n = 2x and then reorder it to get the sum of the squares of the odd integers plus the sum of the squares of the even integers, that is
so
Now apply the expression for the sum of all the integers from 1 to n to get
I simplified this expression to get
Penny |