Sequences - Pascal's triangle

See the resources page for Pascal's triangle.
 

Find the following sequences in Pascal's triangle:

• natural numbers
• triangular numbers
• tetrahedral numbers
• pentatope numbers

Hexagonal numbers are found in Pascal's triangle.

• On triangle paper draw the first few hexagonal numbers.
• Make a table. Describe and explain any patterns.
• Find and explain the relationship between hexagonal and triangular numbers?

Investigate Fibonacci numbers in Pascal's triangle.
 

For triangular, tetrahedral, pentatope and hexagonal numbers, find the formula for the general term as a combination and simplify.
 

e.g. n-th triangular number = ⁿ⁺¹C₂ = ... = ½n(n+1)

The formulas are available from Math Forum.