Check to see whether the difference between terms is a constant, such as in the sequence 20, 40, 60, 80, 100.... If it is, then this is an arithmetic sequence that follows the formula f(n) = f(1) + (n-1)d, where f(1) is the first term, d is the common difference, and n stands for the number of terms. In our example:
f(1) = 20
d = 20
f(n) = 20 + (n-1)20 = 20 + 20n - 20 = 20n
Hence, the pattern for this number sequence is: f(n) = 20n
The 10th term would be f(10) = 20 x 10 = 200
Check to see whether the second of two consecutive terms can be divided by the first. If there is such a common ratio r, then this is a geometric sequence. For example, in the sequence 15, 45, 135, 405, the common ratio r = 3 since
405/135 = 135/45 = 45/15 = 3
If this is a geometric series, construct the formula: f(n) = f(1) x r^(n-1)
In our example:
f(1) = 15
r = 3
f(5) = 15 x 3^(5-1) = 15 x 3^4 = 15x8 = 1,215
Compare common differences of successive terms. List the numbers in your pattern in line 1. List the differences of these numbers in line 2 below line 1. List the differences between the numbers of line 2 in line 3. Continue this method and observe whether a pattern appears. For example:
Line 1: 2, 5, 10, 17
Line 2: 3, 5, 7, 9
A pattern of common differences appears in line 2.
Compare common ratios of successive terms. List the numbers in your pattern on line 1. List the ratios of these terms on line 2 below line 1. Continue this method and observe whether a pattern appears.
Check for other common mathematical operations and patterns in the terms. For example, check for sequence of squares, sequence of cubes, sequence of fourth powers and sequence of factorials.
Check for other common mathematical operations and patterns in the successive terms. Check for sequence of squares, sequence of cubes, sequence of fourth powers and sequence of factorials in the successive terms.
Check for famous sequences. The math forum and the on-line Encyclopedia of Integer Sequences delineate formulas for many famous sequences. Two examples are the formulas for Fibonacci sequence and the formula for triangular sequence.
Fibonacci sequence: F(n) = (a^n - b^n)/(a - b), where a and b are the roots of the quadratic equation x^2-x-1 = 0.
Triangular numbers sequence: F(n) = [n(n + 1)]/2