Question

antoni is designing an installation for his town square. the installation will be a pyramid in which each level is a solid layer of identical blocks. the diagram below shows the first 3 levels. if antonis design calls for the pyramid to be 6 blocks tall, how many total blocks will he need for the whole installation?

Explanation:

Step1: Observe the pattern

The number of blocks in each level forms a sequence of square - numbers. The first level has $1^2 = 1$ block, the second level has $2^2=4$ blocks, the third level has $3^2 = 9$ blocks. In general, the $n$th level has $n^2$ blocks.

Step2: Calculate the sum of square - numbers

We need to find the sum of the squares of the first 6 positive integers. The sum of the squares of the first $n$ positive integers is given by the formula $S_n=\sum_{k = 1}^{n}k^2=\frac{n(n + 1)(2n + 1)}{6}$.

Step3: Substitute $n = 6$ into the formula

When $n = 6$, we have $S_6=\frac{6\times(6 + 1)\times(2\times6+ 1)}{6}$.
First, calculate the values inside the parentheses: $6+1 = 7$ and $2\times6 + 1=13$.
Then, $S_6=7\times13=91$.

Answer:

91