you are creating a window display at a toy store using wooden blocks. the display involves stacking blocks in triangular forms. you begin to display with 1 block, which is your first "triangle," and then stack three blocks, two on the bottom and one on the top, to get the next triangle. you create the next three triangles by stacking 6 blocks, then 10 blocks, and then 15 blocks. how many blocks will you need for the ninth triangle?
Your numbers are called "triangular" numbers and are found in Pascal's triangle.
see:
http://milan.milanovic.org/math/english/triangular/triangular.html
To determine how many blocks you will need for the ninth triangle, we can look for a pattern in the number of blocks used for each triangle.
From the given information, we know that the first triangle uses 1 block, the second triangle uses 3 blocks, the third triangle uses 6 blocks, and the fourth triangle uses 10 blocks.
Let's observe the differences between the consecutive numbers of blocks used for each triangle:
- The difference between the second and first triangle is 3-1=2 blocks.
- The difference between the third and second triangle is 6-3=3 blocks.
- The difference between the fourth and third triangle is 10-6=4 blocks.
Notice that the differences between consecutive triangles form a pattern: 2, 3, 4.
To find the number of blocks for the ninth triangle, we can continue this pattern:
- The difference between the fifth and fourth triangle would be 4+1=5 blocks.
- The difference between the sixth and fifth triangle would be 5+1=6 blocks.
- The difference between the seventh and sixth triangle would be 6+1=7 blocks.
- The difference between the eighth and seventh triangle would be 7+1=8 blocks.
Now we can find the number of blocks for the ninth triangle by adding the difference between the eighth and seventh triangle (8 blocks) to the number of blocks used for the eighth triangle:
10 blocks (eighth triangle) + 8 blocks (difference) = 18 blocks
Therefore, you will need 18 blocks for the ninth triangle.