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.