Question

(a) How many blocks are there in the solid figure shown?
blocks

(b) How many blocks are there in a similar figure with 50 layers?
blocks

Answer 1

you have 5 layers in that solid figure. Each layer is a square. The bottom most layer has 5 blocks on the side. Therefore, it is a square of side 5. To count the number of blocks in that layer, you simply have to find the area of that square. So, the number of blocks in the bottom layer is area of square with side 5 or 5^2 or 25.
similarly, the layer just above the bottom layer is a square of side 4 and so on till you reach the top.
So, the total number of blocks is: 5^2+4^2+3^2+2^2+1^2.

For a similiar figure with n layers, the formula is
\[1^{2}+2^{2}+3^{2}+.......+..(n-1)^{2}+n ^{2}\]
for 50 layers, n = 50