Let's make a little diagram that we can check all of our answers with.
Month Cars Made This month Cares Made to Date 1 1 1 2 2 3 3 3 6 4 4 10 5 5 15 ... ... ... n an Sn
A. Following the pattern, Sn = Sn-1+an. Yet, an = n, so we have that the number of cars made to date in the nth month is Sn = Sn-1+n. But Sn-1 = Sn-2 + n-1, so Sn = Sn-2 +(n-1) + n. Thus, continuing its recursive definition, Sn = Sn-3 + (n-2) + (n-1) + n. Continuing this until we get down to the first month, this is Sn = 1 + 2 + ... + n, which can be shown to equal (by the sum of an arithmetic sequence):
With a quick check of out table, we can see that this gives us the third entry in every row.
B. After the first year (n = 12), we can either substitute n=12 into the formula above or do 1 + 2 + ... + 12. Either way, we get 78 cars produced.