Questions based on number series are a favourite with most examiners. In such a question, if it consists of a single series of numbers, they are arranged in a particular manner and each number is related to the earlier and the following numbers in a particular way. The examinee is supposed to decode the logic involved in the sequence and then fill in the space containing the question mark with a suitable choice out of those given. But before we proceed to discuss the various types of questions related to them, we will talk of some basic facts which are essential to an understanding of these types of questions.

Consider the following example, which can be labelled as one of the easiest one can think of.

Quite often, you can find the right answer in number series by taking the difference between consecutive pairs of numbers, which form a logical series. As in the above example, the differences between succeeding pairs of numbers are 3, 5, 7,9, 11, 13. Logically speaking, the next difference must be 15. This 15, if added to the last number 49, gives you 64, which is the correct answer. But thinking from a different viewpoint, all these numbers given in the questions are squares of natural numbers 1, 2, 3, 4, 5, 6, 7. So the next number in the series should be the square of 8, which is 64. So we can get the same correct answer in series, even if we adopt two different logical approaches.

An important conclusion emerges from this example, i.e., if we remember squares upto a reasonable limit, say 32 and cubes upto 30, we can get a lot of help in solving such questions. Decoding the logic in a series becomes quite easy, if one has an idea of different squares and cubes. Overall, finding the differences between two consecutive numbers or two alternate numbers (if there are two parallel series running) remains the most helpful approach to solving such questions.

Now examine the following series:

The above series involves two operations, multiplication and addition of a number by the same number. The series runs like this 1x1+1=2, 2x2+2=6, 6x3+3=21,21x4+4=88, 88x5+5=445. The next number, following this logic, should be445x6+6=2676. This is one case where the idea of differences will not help us.

Many number series are based on squares, cubes and addition/subtractions to/from them. Have a look at the following case.

### EXAMPLE

Obviously,the first differences do not appear to contain any logic. But if we take the differences among differences as illustrated above, the logic becomes clear. By the same logic, the last addition to the last number i.e. 30, should be 6,which makes it 36. Moving backwards, 36+90+210=336, which is the correct answer. But this method proves to be time-consuming.

Adopting a more sophisticated approach, these numbers move in this fashion.

1

^{3}- 1, 2^{3}- 2, 3^{3 }- 3, 4^{3 }- 4, 5^{3 }- 5, 6^{3 }- 6. Following the same logic, the next number in the series has to be 7^{3 }- 7 =336.**Example No. 4**

**:**3,5, 7, 11, 13, 17, ?

**Solution**

**:**This is a series of prime numbers, it can be clearly seen that every number given isprime in the same numerical order. The next prime number would be 23, the right answer.

**Example No. 5**: 4, 6, 10, 16, 24, 34, ?, 60

**Solution**

**:I**n this case the addition to the numbers increases by 2 i.e. first 2 is added, then 4, then 6and so on. It can be seen that before the questions mark 10 has been added,therefore, the next addition would be 12. And the number would be 34 + 12 = 46.

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