Sometimes, it is possible to simply look at a number pattern, recognize what is happening, and figure what number should comes next. In other cases, when the sequence is more complex, it’s not as easy to decide how it was created. When you encounter these more complicated patters, it is helpful to have a strategy for finding how the pattern was mathematically determined. Once you know how to find the pattern, you can find any number in the sequence.
How to Solve a Number Pattern
Determine if the mathematical distance between the numbers is the same by subtracting each number from the number that follows it. Start by subtracting the first term from the second, and then subtract the second term from the third, until you have checked the distance between all the terms of the sequence. If the distance is the same, you have solved the pattern. If it is not, go on to step 2.
Look for a pattern in the differences between the numbers you found in Step 1. You may find that they get larger by a certain number each time: for example, they could be 1, 3, 5, 7, 9. If there is no obvious pattern in the differences, go on to Step 3.
Return your attention to the original number pattern, and look for a common denominator. For example, if the pattern is 3, 9, 15, 21 ... The common denominator is 3; if we divide by this common denominator, we discover the pattern is 3 times the odd numbers on the number line.
If you still have not found a solution, look for a pattern in the numbers as they are written. This means that instead of looking at a mathematical solution, you look for a code. For example, you might be given the following sequence: 1, 12, 121, 1213, 12131. Here the next number, 121314, is in the pattern of digits as they are written, not in the way they are mathematically manipulated.
If you complete Steps 1-4 without success, begin back at Step 1, taking careful consideration of each step. This should yield a solution.
TL;DR (Too Long; Didn't Read)
Extend the pattern out two or three numbers further than the original sample you were given. See if the rule you created in solving the pattern holds. This is an excellent way to check your answer.
It is easy to become distracted by frustration. If you find yourself becoming too discouraged to calmly approach the problem, take a 10-minute break and look at it with a fresh eye.