You decide to build a pyramid out of wooden blocks. Each block must rest on the corners off 4 other blocks. If there is one block on top and 36 on the bottom, what is the total number of blocks you will need to complete the pyramid?
The top layer has 1 block
the 2nd layer from the top has 4 blocks
the 3rd layer from the top has 9 blocks
the 4th has 16 blocks
the 5th has 25 blocks, and
the 6th has s36 blocks
So we have 1+4+9+16+25+36
= 91
btw, there is a formula for the sum of squares of the first n numbers
sumofsquares(n) = n(n+1)2n+1)/6
sumofsquares(6) = 6(7)(13)/6 = 91
To find the total number of blocks needed to complete the pyramid, we need to calculate the total number of blocks in each layer and then sum them up.
Let's break down the problem step by step:
1. The top layer of the pyramid has only one block.
2. The second layer directly below the top layer has 4 blocks in a square formation.
3. The third layer has 9 blocks arranged in a larger square.
4. The fourth layer has 16 blocks arranged in an even larger square.
5. And so on...
We can observe that each subsequent layer of the pyramid contains one more block on each side. This pattern continues until we reach the 36th layer, which has 36 blocks arranged in a square.
To find the total number of blocks in each layer, we can use the formula for the sum of the first n squares:
Sum = n(n + 1)(2n + 1) / 6
For the 36th layer, we can substitute n = 36 into the formula:
Sum = 36(36 + 1)(2 * 36 + 1) / 6
Simplifying this equation, we get:
Sum = 36 * 37 * 73 / 6 = 37 * 73 * 6
So, the total number of blocks in the 36th layer is 37 * 73 * 6.
To find the total number of blocks needed to complete the pyramid, we need to sum up the number of blocks in each layer from the top to the bottom. This can be done by finding the sum of the arithmetic sequence:
Total = [(Number of blocks in first layer + Number of blocks in last layer) × Number of layers] / 2
Let's calculate the total number of blocks:
Total = [1 + (37 * 73 * 6)] * 36 / 2
Simplifying this equation, we get:
Total = [1 + (37 * 73 * 6)] * 18 = 27054
Therefore, to complete the pyramid, you will need a total of 27,054 blocks.