Question

What can you say about the relationship between the row number and the second term in each row in pascal's triangle?

Answer 1

1
1 1
1 2 1 and so on.

First row: 1. This 1 represents any nonzero base raised to power 0: b^0=1.

Second row: 1 1. These are the coefficients of a first order expression such as 1x^1 + 1x^0, or x+1.

Third row: 1 2 1 These are the coeff. of a 2nd order expression such as
1x^2 + 2x + 1 = (x+1)^2

So it appears that the nth row contains the coefficients applying to the (n-1)th power of a binomial, such as (a+b)^n.

If n = 3, then the 3rd row 1 2 1 contains the appropriate coeff. for the expansion of (a+b)^(3-1) = (a+b)^2 = a^2 + 2ab + b^2.