Basic Recurrence Relation Problem

A factory makes custom sports cars at an increasing rate. In the first month only one car
is made, in the second month two cars are made, and so on, with n cars made in the nth
month.
(a) Set up a recurrence relation for the number of cars produced in the first n months by
this factory.
(b) How many cars are produced in the first year?

a. I think this would be a_{n =} a_n-1 + 1, but something tells me it might be a_{n =} a_n-1 + n, I am not sure.
b. Would this just be 1+2+3+4+5+6+7+8+9+10+11+12? Or, plugging in 12 for the above equation?

Thanks.

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	an	Sn

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.