Write the numbers that begin the sequence on a piece of paper that you can write on.
Examine the full sequence to see if there is an obvious pattern. For example, in the following number sequence each number increases by two: 2, 4, 6, 8, 10.
Draw a peaked line connecting each number with its neighbor if there is no obvious pattern.
Write the rule that you could apply to the first number to get to the second above each peak. If there are multiple options, start with the simplest.
For example, in the number sequence 15, 20, 26, 34, 45, 60, you could add 5 to 15 to get 20 or multiply 15 by 1 1/3. Adding 5 is the simpler answer, so start with that. Using this rule of addition, you get a second set of numbers above the first: +5, +6, +8, +11, +15.
Examine the second set of numbers for any obvious pattern. If there is a pattern, such as +1, +2, +3, you have solved the problem. If not, continue to step six.
Draw peaked lines connecting each number in the second level with its neighbor.
Write the rule that you could apply to the first number to get to the second above each peak. If a pattern emerges, you have solved the problem. If not, continue to step 9.
For example, in the number sequence +5, +6, +8, +11, +15, you can add 1 to 5 to get 6. Likewise, you can add 2 to 6 to get 8. A pattern has emerged: +1, +2, +3, +4.
Extrapolate the next number in the sequence using the patterns.
For example, the next number in the top sequence is +5, which makes the next number in the second layer +20, which makes the next number in the original sequence 80.
Try a different method if no pattern emerges after you have looked for one in multiple tiers. For example, if you find no number pattern using addition, try multiplication or exponents.