Hi Walshy,

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 a _{n}S _{n}

A. Following the pattern, S_{n}= S_{n-1}+a_{n}. Yet, a_{n}= n, so we have that the number of cars made to date in the nth month is S_{n}= S_{n-1}+n. But S_{n-1}= S_{n-2}+ n-1, so S_{n}= S_{n-2}+(n-1) + n. Thus, continuing its recursive definition, S_{n}= S_{n-3}+ (n-2) + (n-1) + n. Continuing this until we get down to the first month, this is S_{n}= 1 + 2 + ... + n, which can be shown to equal (by the sum of an arithmetic sequence):

S_{n}=

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.