As has been discussed, sequence problems generally involve finding patterns among the items in a sequence, or the defining rule for the sequence. Generally, for questions involving the sequence items themselves, the best approach involves writing down information (often in the form of an equation) for specific items in the sequence, and trying to find a pattern among these items.
If Sn = 3n, what is the units digit of S65?
Clearly, you cannot be expected to multiply out 365 on the GRE, even with a calculator. Therefore, you must look for a pattern in the powers of three.
You can see that the units digits of powers of 3 follow the pattern “3, 9, 7, 1” before repeating. The units digit of 365 will thus be 3, because the 64th term will be “1” as 64 is divisible by 4 (and the pattern repeats every four terms).
As a side note, most sequences on the GRE are defined for integers n ≥ 1. That is, sequence Sn almost always starts at S1. Occasionally, a sequence might start at S0, but in that case, you will be told that n could equal 0.
If An = 7n − 1, what is the units digit of A33?