This is the original problem that my question is coming out of:
A number X is chosen at random from the series 2,5,8,... and another number Y is chosen at random from the series 3, 7, 11, ... Each series has 100 terms. Find P[ X=Y ].
The answer to this problem is to see that 100^2 = 10,000 equally likely possible choices of (X,Y). Of these choices, the pairs that equal X and Y are (11,11), (23,23), (35,35), (299,299), etc. There are 25 such pairs, so the probability is 25/10,000.
My question is what is the proper mathematical way to determine those pairs of (X = Y)? The answer key tells you that they are of the form (12k - 1, 12k - 1). Working backwards from that, I see that since the formulas for the two sequences are:
2 + 3(n - 1),
3 + 4(n - 1),
respectively, you can multiply each out to get:
3n - 1,
4n - 1.
Is finding the common terms as simple as just multiplying the coefficients of n for each? Would it matter if their constants weren't equal, unlike in this case?
This isn't touched on directly in the book (it's for the Actuarial P/1 Exam), and I haven't been able to find anything on the internet about finding common terms of sequences.