arithmetic sequence help?

Here is how I would work this.  I will start with the general form for the n'th term of an arithmetic sequence:

a(n) = a + b(n - 1)

Where a is the first term and b is the common difference.  At this time, we know neither.

We are told that the 20th term is 20.  We can substitute that information into the above to get an equation:

a(20) = a + b(20 - 1)

20 = a + 19b

We'll leave that alone for now.  Next, we'll look at the equation for the sum of the first n terms of an arithmetic sequence:

S(n) = [ a + a(n) ] n / 2

We don't know what a is, but we do have an expression for a(n).  I'll substitute that here and simplify what I can:

S(n) = [ a + a + b(n - 1) ] n / 2

S(n) = [ 2a + b(n - 1) ] n / 2

We are told that the sum of the first 20 terms is 100.  We can substitute those into the above to get a second equation:

S(20) = [ 2a + b(20 - 1) ] * 20 / 2

100 = (2a + 19b) * 10

10 = 2a + 19b

We now have a system of two equations and two unknowns.  You are asked to solve for the common difference (b), so I'll start with solving the first equation for "a" in terms of b so we can substitute it into the second equation:

20 = a + 19b

20 - 19b = a

10 = 2a + 19b

10 = 2(20 - 19b) + 19b

10 = 40 - 38b + 19b

-30 =-19b

30/19 = b

The answer to the question is 30/19 (option B).  I'll stop here since we've answered the question but if you wanted to solve for "a" to see what the first term was, you can do that using one of the two equations above and the value of "b" that we just calculated.