What should be the general approach to solve the next number of a series?

Question

As a job-seeker, I often have to face quantiative aptitude examination before getting entrance to interviews. A common problem found in the quantiative aptitude examination is to determine the next element of the given series; simple yet tricky. So what should be our step-by-step approach to solve them? A quick Google search of the series may give us a particular solution to a particular problem (sometime does not). But I believe there is a way to approach and solve this type of questions. Otherwise sometimes it becomes impossible to solve some serieses. So how should we proceed to solve them?

Some Examples:

1: $\{4, 2 ,36, 4, 5, 1, 36, 16, 6, 3, 81, \ldots\}$

2: $\{\frac{1}{4},\frac{1}{4},\frac{1}{2},\frac{3}{2},6,\ldots\}$

3: $\{7,9,11,6,11,8,5,13,5,4,15,\ldots\}$

4: $\{2,1,2,3,2,9,9,0,1,1,9,\ldots\}$

5: $\{2, 1, 2, 3, 8, 0, 1, 9, 1, 2, 1,\ldots\}$

6: $\{4,2,5,9,5,11,13,7,16,17,9,\ldots\}$

NB- I don't want to know solution to the mentioned serieses. Even I can get them from Facebook Groups. I just want to know how to approache them. Thank you.

Answer 1

There is no definitive answer to this question. However, here are some approaches I use:

1. Write all fractions in non-reduced/mixed number form--look for patterns in the numerator and in the denominator.
2. Look for what the numbers have in common (in general); if they're all perfect squares, etc, that can provide some guidance on where to look.
3. What numbers appear often? Is there a pattern to these numbers?
4. Does the sequence appear to converge to a value?
5. Know what operators are allowed. For example, $\{0, 1, 0, 1, 0, 1, 0, ...\}$ is simple to write if the modulo operator is legal: $a_n = n\mod 2, \text{for } n\ge0$, but slightly harder if it is not: $\frac{(-1)^{n+1} + 1}{2}$.
6. Practice! :) This is the only guaranteed way to get better at these sorts of things.

Answer 2

There is a reason why they put questions that are not very technical but challenging. These problems test how quickly you analyze patterns, or rather how agile your brain is. The best answer is "look for pattern" or "practice lot of them". Observe how each new number in the sequence varies. Look for rules. Try, arithmetic operations, or "string patterns", recursion etc. Try to think all the "rules" that might have been used. Obtain a new sequence by substraction consequitive terms. e.g. if you have $a_1,a_2,a_3,a_4,..$, then make new sequence $a_2-a_1,a_3-a_2,..$. Try for division. Does a pattern you knew appear? If pattern does not appear after one or two or three numbers, it may appear after four terms.

Learn famous sequences like fibonacci, sum of first n even numbers, and other common sequences, primes, perfect numbers, and so on.

This is all, I know. And you also knew. But I don't know more than that.

UPDATED

Out of Curiosity are these correct? $(1)64,7,2,81,...(2)30,180,1260...(3)2,3,17,...$