When solving math problems, it is often good if we are able to recognize patterns in the numbers that we are seeing. This will allow us to hypothesize what the general term of the sequence will look like, which could help to guide the approach for the problem.
You should be aware of sequences like the integers, odd numbers, perfect squares, primes, factorials, exponents, etc, and these should be easily identified.
Trying to determine the pattern from a sequence of numbers can be considered an art. There is no strict rules that we can follow which will guarantee a result. We can merely use our powers of observation and intuition to guide our guesses. Some common approaches are to look at successive differences between terms, adding another sequence (often constant or linear) to the current numbers, dividing out by any common factors, etc.
The On-Line Encyclopedia of Integer Sequences is a good reference that you could use, if you are confronted with a mathematical sequence and are unable to recognize the pattern. It is the largest database of integer sequences, storing over 220, 000 sequences that are of interest to mathematicians and amateurs. It was started by Neil Sloane as a graduate student in 1965, to help him recognize sequences that arose from his work in combinatorics.
1. What comes next: \( 1, 3, 5, 7, \_ \)?
Solution: We recognize this as the list of odd numbers, and so the next term is 9. \( _\square \)
2. What comes next: \( 2, 3, 5, 7, \_ \)?
Solution: We recognize this as the list of primes, and so the next term is 11. \( _\square \)