# Number Series: Fundamentals

When it comes to number series questions, a series of numbers is given with a number missing. This missing number can be anywhere in the sequence. There can also be questions where one of the numbers in the series is wrong. One is supposed to find a consistent logic in the given series and answer accordingly.

In this article, we will go through number series questions. I have also added my solutions to these questions. Through these questions, we will look at various methods of solving the same question (wherever possible) and understand what’s best in the exam situation. For these questions, I have not provided options so that you develop the ability to crack these questions without any support/guesswork using options.

Q.1 – 7, 12, 19, ?, 39

Solution: Take the difference between consecutive terms which is 5, 7, and so on. Which means that either it is odd number difference or prime number difference. If it is odd, we will get 19 + 9 = 28, and then 28 + 11 = 39. Satisfies. It won’t be prime as 19 + 11 will give us 30 and then the next difference will become 39 – 30 = 9. Hence, the logic here is odd number difference and the next term will be 19 + 9 = 28.

Another logic which a lot of people won’t be able to crack is the n^2 – something logic. Here,
7 = 3^2 – 2
12 = 4^2 – 4
19 = 5^2 – 6
missing term
39 = 7^2 – 10.

Missing term will be 6^2 – 8 = 28. Lengthy approach for sure and this typically won’t strike during the test.

Q.2 – 0, 6, 24, 60, 120, 210, ?

Solution: Most of the times, you will encounter some common numbers where it is important to remember the logic. For example, 210 which is nothing but 6^3 – 6. That gives away the logic of this question that it is n^3 – n. So the sequence is:

1^3 – 1, 2^3 – 2, 3^3 – 3, 4^4 – 4, 5^5 – 5, 6^3 – 6, 7^3 – 7 = 336 will be the answer. Any alternate logic? yes!

6 = 2*3
24 = 4*6 = 4(3+3)
60 = 6*10 = 6(6+4)
120 = 8*15 = 8(10+5)
210 = 10*21 = 10*(15+6)

Next number will be 12*(21+7) = 12*28 = 336. Slightly lengthy compared to the first approach but gives the solution. Any other method? Let’s try taking the difference between consecutive terms.

6 – 18 – 36 – 60 – 90

6*1, 6*3, 6*6, 6*10, 6*15. The next term will be 6*21 = 126. 210 + 126 = 336!

Q.3 – 4, 6, 12, 14, 28, 30, ?

Solution: Simple. 4 + 2 = 6; 6*2 = 12; 12 + 2 = 14; 14*2 = 28; 28 + 2 = 30. The next term will be 30*2 = 60

Q.4 – 1, 3, 3, 6, 7, 9, ?, 12, 21.

Solution: Here, once you try the first difference between terms, you will get 2, 0, 3, 1, 2, and so on which doesn’t make much sense. In such cases, it is advisable to check for alternate terms. We will find two sequences here. 1, 3, 7, ?, 21 and 3, 6, 9, 12. As we need to tackle the first one with the missing number, the second sequence doesn’t really matter. 1, 3, 7, ?, 21 will give us term differences as 2, 4, 6, 8 and hence, the term will be 7 + 6 = 13.

Q.5 – 1, 2, 5, 26, ?

Solution: It is easier to figure out 5 = 5^2 + 1 or 5*5 + 1 than starting from 1-2. If we work the logic backwards, we will get it as: 1^2 + 1 = 2; 2^2 + 1 = 5; 5^2 + 1 = 26; and the next term will be 26^2 + 1 = 677.

Q.6 – 96, 6, 48, 18, 24, 54, ?

Solution: In this question, one can always start from 96/16 = 6 and so on but when I observe 6, 18, 54 the logic starts materializing. These are two alternate sequences. Hence, 96 -> 48 -> 24. The next will be half of this = 12. Taking difference as the first attempt at cracking the logic is fair, but since it is ridiculously common, you won’t find it in a lot of moderate to difficult level questions.

Q.7 – 1, 10, 66, 469, ?

Solution: When the difference between two numbers increases substantially, the logic is generally related to multiplications and powers. So, I will start with 66 and 469. 469 = 7*67 which gives me a starting point that it is somehow related to 66. So 66*7 = 462 + 7 = 469. Working backwards, this is what I can establish:

1*5 + 5 = 10
10*6 + 6 = 66
66*7 + 7 = 469
469*8 + 8 = 3760

Q.8 – 2, 17, 101, 362, ?

Solution: Where did 362 come from? Oh, I know! It is close to 361 which is 19^2. Can the logic be squares + or – something? Absolutely.

1^2 + 1 = 2
4^2 + 1 = 17
10^2 + 1 = 101
19^2 + 1 = 362

What’s the logic behind 1-4-10-19? The difference is increasing by 3-6-9. So the next term will be 19 + 12 = 31. 31^2 + 1 will be 961 + 1 = 962

Q.9 – 15, 31, 95, 239, ?, 895, 1471

Solution: If I take difference in consecutive terms, we will get 16 – 64 – 144 – x – y – 576 and these are squares of 4, 8, 12 and the ones in between should be 16^2 = 256 and 20^ = 400 to give us 24^2 after 895. So 239 + 256 = 495 will be the answer. One might start with 15, 15*2 + 1 = 31, 31*3 + 2 = 95, but that doesn’t help. If you started with this logic, you should stop here and think of alternate method.

Q.10 – 1, 2, 8, 33, 148, ?

Solution: If we take the difference, we get 1, 6, 25, 115 which I don’t think is taking us anywhere. As the difference is increasing significantly after 33, the logic can be into something + or – something.

1*1 + 1 = 2
2*2 + 4 = 8
8*3 + 9 = 33
33*4 + 16 = 148.

The next term will be 148*5 + 25 = 740 + 25 = 765.

Hope you got a fair idea of how to approach these questions. To master number series questions, you need a strong command over tables, squares, cubes, exponents, and a knack of identifying pattern by splitting numbers. Typically, if you apply 5/6 initial approaches (difference, alternate series, difference of difference, into something + something, and powers), you will make some breakthrough. However, it is always possible that the logic is completely arbitrary. Spend a minute or so. If you still don’t get it, make an intelligent guess and move on.

If there are any queries, please leave them in the comments below and allow me a few days to get back. Share this with your friends and co-aspirants. Happy prepping! 🙂

## 2 thoughts on “Number Series: Fundamentals”

1. Jugal

Hello sir!
1. PS