Can someone explain pascal's triangle to me. I found it in a math contest.

How do you find the sum of numbers from 1-n. There is a question that wants me to find the sum from 1 to 101.

When trying to solve a problem and working a long time on it, do you ever worry that it requires mathematics from another field that you never studied before? How do mathematicians know that the problem is contained within their mathematical limits and experience and training. What are some famous examples of problems that were solved by relating it to different areas of mathematics?