List the first four numbers in the sequence and ignore the rest. For example, if you are given the sequence of numbers "8, 12, 16, 20, 24, 28, 32, 36" and so on, do not worry about any of the latter numbers in the sequence. For most sequences, knowing the first four is enough. Thus, in this example, you would only observe the numbers 8, 12, 16 and 20.
Observe the differences between sequential numbers. For our example, we might immediately notice that the sequence is increasing by four each time. If this is not clear immediately, subtract a number in the sequence from the next number. Specifically for the example, you can see that 12 -- 8 = 4, 16 -- 12 = 4 and 20 -- 16 = 4. In other words, there is a difference of four between each set of numbers in the sequence.
Find what is common in terms of change in the sequence and use a variable (known as the subscript) to represent that common change. Usually in mathematics, the variable "n" is used to represent the index of the sequence. For instance, when you use "n" in a sequence with subscripts, you can plug in any integer for "n" to find the number with that index. In our example, we know that since the number in the sequence increases by four for each step in the index (each increase of one in "n"), you should multiply the variable by 4. Thus, we know that 4n must be in the representation.
Replace "n" with "n-1." This step is necessary in that it shows that the first number in the sequence has not changed. In other words, while in many mathematical situations, the first number in a set is labeled as zero, in sequences this is not the case and must be corrected. In the example, write 4(n -- 1) to replace 4n.
Add the beginning value to the representation of the change in the sequence. For the example, the beginning value in the sequence is 8; thus you would write 8 + 4(n - 1). This is enough -- it is a sequence with subscripts.
Simplify the sequence with subscripts, if needed. This step is not necessary, but can help make the sequence easier to understand and calculate. In our example, 8 + 4(n -- 1), you can multiply the four that is multiplying (n -- 1) to yield two terms. Doing this yields 8 + 4n -- 4, which further simplifies to 4 -- 4n. This is a simplification of the sequence.