how to find numbers in a sequence

Look for a pattern in the series of numbers by looking at the differences between terms. For example, in the series 1, 2, 4, 7, 11, the differences are 1, 2, 3 and 4. There are specific intervals between the numbers--1+1=2 2+2=4 4+3=7 7+4=11. The next number in the sequence would be 11 + 5 = 6. The differences could be something less obvious, like prime numbers or squared numbers. When looking for a pattern consider these as a starting point:

1) 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.

2) Check to see if 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

3) 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.

4) Check for other common mathematical operations and patterns. For example, check for sequence of squares, sequence of cubes, sequence of fourth powers and sequence of factorials.