============= Introduction ============= Number Sequences ============= Number Series ============= Fibonacci Sequence ============= The Ultimate Triangle ============= Credits ============= |
## Garden Last: Pascal's Palace- Pascal's Triangle
Hush friends! We are about to enter a palace--a pure-white palace built entirely out of white marble bricks. The architecture's Gothic style is absolutely breath taking; yet there are more to expect inside! Here is the key to the gate of our last garden--we're being greeted by a tall "pyramid"--or triangle, rather, made of white marble cubes. This is a pattern! ...Let's do something with this pyramid. Starting with 1 at the top,
we'll fill in the pyramid What? Too much? Not at all! What have you noticed? A We've filled in the entire triangle now! But we're just starting. This triangle is
called the - Garden Highlights
**Diagonal Rows 1, 2 and 3**
Row 1 down both sides of the triangle from the top is straightforward--they're all 1's! Row 2 is the sequence of counting numbers or Row 3: 1,3,6,10,15,21,... sounding familiar?? Yes, THE TRIANGULAR NUMBERS!!! **Horizontal Rows**
Check out the following: If we add up all of the numbers in a horizontal row starting at row 2, the answer would always be equal to the power of 2. **"The Fib"**
Let's try some more additions... We'll start with the first square:
{ 1, 1, 2, 3, 5, 8, 13, 21, ... } There's the Fibonacci Sequence in the Pascal Triangle! **Triangles within Triangle**
Find all of the even numbers in the triangle and highlight them: They all form upside-down triangles (the single-square can symbolize a point, which in turn can
symbolize a very small triangle)! Hence, the number of cubes in each of these small
triangles is triangular numbers!
Tadaaa! More sets of triangles!!! You can also try the divisibles by 3 and 4 and check out the patterns they create on the triangle! |

© 2009 Grace HY Ma |