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Sequences and series (as the name suggests) consists of questions where you are supposed to understand the logic behind a given sequence or series of number/alphabets. Based on this understanding you are supposed to determine either:
A continuation to the series
The immediate next term of the series or
A missing term/terms within the series.
While preparing yourself to solve such questions, you should improve your ability to spot a particular relationship between terms within the series. The common logical premises used for questions are given here.
Introduction
Number series is a form of sequence, where some numbers are mistakenly put into the series of numbers and some number is missing in that series, we need to observe first and then find the accurate number to that series.
Sequence is the list of numbers written in the specific order.
For ex. 1, 4, 9, 16, 25, 36……..
Some Common types and Their Tricks
Type 1 - Addition / Subtraction or Multiplication / Division Example 1: 19, 23, 39, 75, _, 239
Most common trick to solve a number series is to solve by checking difference between two adjacent numbers, but difference could not only lead to addition/subtraction but it can also be with multiplication/division. Therefore, to check whether to think addition wise or multiplication wise in increasing sequence one must make assumption with the help of difference between first and last number of the given sequence.
If the difference seems to be less according to the number of steps used to make last number from first than we should check addition
If the difference seems to be large or too large one must check multiplication trick between the adjacent numbers
In the Exercise 1. given above, difference between first no. (19) and last number (239) is 220. Now a question will arise how we will assume whether the difference is more or less. We will assume it by keeping in mind number of steps required to start from first number till last number. In Example 1, 19 becomes 239 in five steps, because there are four more numbers between them, one of which we have to find out. Difference of 220 between 19 and 239 in five steps logically giving priority to addition over multiplication in an increasing sequence like this. 75 + 64 = 139, 139 + 100 = 239 139 is correct answer.
Question for Number Series Notes - 1
Try yourself:Look at this series: 7, 10, 8, 11, 9, 12, ... What number should come next?
Explanation
This is a simple alternating addition and subtraction series. In the first pattern, 3 is added; in the second, 2 is subtracted.
View Solution
Example 2: 10, 31, 95, 288, ___, 2609 As we can see in above ex. Difference between first number (10) and last number (2609) is 2599 in five steps, which indicate us to check multiplication trick between the numbers. (288*3) + 4 = 868, is the correct answer. Note: While checking multiplication trick always start from right end of the sequence.
Example 3: 30, 34, 43, 59, 84, 120,? (a) 169 (b) 148 (c) 153 (d) 176 (e) None of these Ans. (a) Solution: The given pattern is: +22, 32, +42, + 62, +72 So, missing term is 169 = 120 + 72
Example 4: 40, 54, 82, ?, 180 ,250 (a) 142 (b) 124 (c) 136 (d) 163 (e) None of these Ans. (b) Solution: The pattern is: +14, + 28, + 42, + 52, + 70 So, missing term is 82 + 42 = 124
Question for Number Series Notes - 1
Try yourself:Look carefully for the pattern, and then choose which pair of numbers comes next - 8,11, 21, 15, 18, 21, 22
Explanation
This is an alternating addition series, with a random number, 21, interpolated as every third number. The addition series alternates between adding 3 and adding 4. The number 21 appears after each number arrived at by adding 3.
View Solution
Example 5: 0, 1, 3, 8, 18, 35, 264 (a) 62 (b) 35 (c) 18 (d) 8 (e) None of these Ans. (a) Solution: The pattern is +(0^{2} + 1), +(1^{2} + 1), + (2^{2} + 1) ,+ (3^{2} + 1), + (4^{2} + 1), + (5^{2} + 1) So, 264 is wrong and must be replaced by 35 + (5^{2}+1) = 62
Example 6: 1, 9, 125, 49, 729, 121, 2147 (a) 2147 (b) 729 (c) 125 (d) 1 (e) None of these Ans. (a)
Example 7: 5531, 5506, 5425, 5304, 5135, 4910, 4621 (a) 5531 (b) 5425 (c) 4621 (d) 5135 (e) 5506 Ans. (a) Solution: The number should be 5555 in place of 5531. -72, -92, -112, -132, -152, -172…
Example 8: 6, 7, 9, 13, 26, 37, 69 (a) 7 (b) 26 (c) 69 (d) 37 (e) 9 Ans. (b) Solution: The number should be 21 in place of 26. +1, +2, +4, +8, +16, +32
Example 9: 1, 3, 10, 36, 152, 760, 4632 (a) 3 (b) 36 (c) 4632 (d) 760 (e) 152 Ans. (d) Solution: The number should be 770 in place of 760. ×1 +2, ×2 +4, ×3 +6, ×4 + 8, ×5 +10, ×6 + 12, …
Example 10: 4, 3, 9, 34, 96, 219, 435 (a) 4 (b) 9 (c) 34 (d) 435 (e) 219 Ans. (d) Solution: The series is 02+ 4, 12 + 2, 32 + 0, 62 - 2, 102 - 4, 152 - 6, 212 – 8… Hence, 435 should be replaced with 433
Example 11: 157.5, 45, 15, 6, 3, 2, 1 (a) 1 (b) 2 (c) 6 (d) 157.5 (e) 45 Ans. (a) Solution: The number should be 2 in place of 1. 3.5, 3, 2.5, 2, 1.5, 1
Type 2 - Perfect Square or Perfect Cube
Question: 4, 18, 48, 100, 180, ___. In case if Type-1 is not applicable in a sequence, then in the next step we must compare given numbers or their differences to square or cube of natural numbers, as in the above example.
Therefore, (7)3-(7)2 = 294, is the correct answer.
1. Perfect Square Series: This Types of Series are based on square of a number which is in same order and one square number is missing in that given series.
Question for Number Series Notes - 1
Try yourself:In the following options, a few number series are present. One of them has an error, pick the wrong one out:
Explanation
Try solving this without looking at the table present above. It takes a lot of time. The first series is just an A.P. with a common difference of 6. Now using the table, we can see that the second sequence is the perfect square sequence. This series can be formed from the series given in option A. The third sequence is a two-tier square series but in place of 841, we must have 900. This is a wrong series. The last series is also a square series. So the correct option or the wrong series of the four options presented above is C) 441, 529, 676, 841
View Solution
Example 1: 841, ?, 2401, 3481, 4761 Ans. 292, 392, 492, 592, 692
Question for Number Series Notes - 1
Try yourself:In the sequence given below, a term is missing. The missing term is written in the options that are present below. Find the missing term and choose it from the options below: 8, 12, 21, 37, __
Explanation
The first term is 8, which is not a square. Although 8 is a perfect cube, the other terms are not. Hence we conclude that this series is neither a perfect square nor a perfect cube series. That means it may be a tier-two series. First, we check the difference between the two beginning terms that is 8 and 12 = 4 or 22. Next the difference between 21 and 12 = 9 = 32. So there is some sort of a pattern. If we could see the same pattern in the next term, we will generalise this rule as the rule of the series.
View Solution
2. Perfect Cube Series: This Types of Series are based on cube of a number which is in same order and one cube number is missing in that given series.
Try yourself:In the following series, a number is such that it does not belong to the arrangement.
8000, 27000, 64000, 105000
Select the option that has this number:
Explanation
Focus on the first digit of the numbers. The first digit in each term is a cube of some number except the last option. 105 isn’t the cube of any number, thus the option to select here is D) 105000. The correct entry should be 125000
View Solution
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