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.
The sequence is the list of numbers written in a specific order.
Example: 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
The most common trick to solve a number series is to solve by checking the difference between two adjacent numbers, but the 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 assume with the help of the difference between the first and last number of the given sequence.
If the difference seems to be less according to the number of steps used to make the last number from first than we should check addition.
If the difference seems to be large or too large one must check the multiplication trick between the adjacent numbers.
In Example 1, given above, the 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 the number of steps required to start from the first number till the 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.
The difference of 220 between 19 and 239 in five steps logically giving priority to addition over multiplication in an increasing sequence like this.
Question 1: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.
Example 2: 10, 31, 95, 288, ___, 2609 As we can see in the above example, the difference between the first number (10) and the 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:
+4, +9, +16, +25 and so on. So, missing term is 169 = 120 + 49
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 2: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.
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 +(02+1), +(12+1), + (22+1) ,+ (32+1), + (42+1), + (52+1) So, 264 is wrong and must be replaced by 35 + (52+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. The pattern is: +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. The pattern is: ×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.
(a) Perfect Square Series: This type of series is based on the square of a number that is in the same order and one square number is missing in that given series.
Question 3: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.
Question 4: 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.
The difference between 37 and 21 = 16 = 42. Thus this is the rule that you have been looking for. Such series are two-tier square series. The last term will be thus obtained by adding 52 = 25 to the last but one term or 37. Hence, the missing term is 37 + 25 = 62 and the correct option is B).
(b) Perfect Cube Series: This type of series is based on the cube of a number that is in the same order and one cube number is missing in that given series. Example 1: 3375, ?, 24389, 46656, 79507 Answer: 153, 223, 293, 363, 433 (Each cube digit added with seven to become next cube number)
Question 5: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.
➢Type 3 - Factorisation / Prime Factorisation
If Type 1 and Type 2 is not applicable in a sequence, one must try to make factors of the numbers in the next step.
Example: 6, 15, 35, 77, 143, __. In the above example, all the previous tricks are not applicable to get an answer. Hence, We will make factors of the given numbers. (2,3,5,7,11,13) all are prime numbers in ascending order. Hence, 13*17 = 221, is the correct answer.
Question 6:Next number in series 31, 41, 47, 59, __?
Explanation
Two adjacent numbers in the sequence are alternate prime numbers. Therefore, 67 is the correct answer.
➢Type 4 - Fibonacci Series
A series in which a number is made by using previous two numbers are called Fibonacci series.
Example: 1, 4, 5, 9, 14, 23, ___ In the above sequence, all the numbers are the sum of the previous two numbers. Therefore, 23+14 = 37, is the correct answer.
➢Type 5 - Sum of Digits
Example 1: In the above sequence, the difference between two numbers is the sum of the digits of the first number. Hence, 89+17 = 106, is the correct answer.
Example 2: All the numbers are multiplied by their sum or added by their sum alternately. Therefore, 11788 + 25 = 11813, is the correct answer.
➢Type 6 - Alternate Pattern Series
When numbers given as a hint in a question are more or when a question asks two numbers of a series or the same number come twice in a series, these all give a hint to alternate pattern series.
Example 1: So, numbers are 12+5 = 17, 53-6 = 47 Ans. 17, 47
Example 2: So, the number is 24+3 = 27 Ans. 27
Question 7:In the given series, a number is missing. The same number is present in the options below. 32, 40, 24, 16, 24, __ Select the missing number.
Explanation
- Let us first detect the rule. We will check the rule on at least 2/3rd of our terms. Now let us see the first term is 32 and the third term is 24. Since the third term is smaller than the first one we can speculate that it has been obtained by a rule that has either subtraction or division involved. Similarly, the second term is got from a rule that has either addition or multiplication in it. - We see that 32 + 8 = 40 and 32 – 8 = 24. Now let us check this rule for one more term. We see that 16 + 8 = 24. - Therefore this works for more than half of the given numbers. Hence the rule is correct. Therefore the missing number in the series is 16 – 8 = 8. The answer is D) 8.
➢Type 7 - Decimal Pattern Series
When the numbers of the sequence are given in the decimal form is decimal pattern series. Example 1: So, the answer is 18 * 0.8 = 14.4
While using bracket pattern we multiply first outside and either add or sub based on given number.
Example 1: 3, 28, 180, ____, 3676 The pattern is : (3+1)*7 = 28 (28+2)*6 =180 (180+3)*5 = 915 (915+4)*4 = 3676 Ans. 915
Example 2: 37, 31, 52, 144, __, 2810 The pattern is : (37-6)*1=31 (31-5)*2=52 (52-4)*3=144 (144-3)*4=564 (564-2)*5=2810 Ans. 564
➢Type 9 - Dual Pattern Series
Example: 15, 9, 8, 12, 36, 170 19, a, b, __, d, e, The pattern is : (15-6)*1=9 (9-5)*2=8 (8-4)*3=12 Similarly: (19-6)*1=13 (13-5)*2=16 (16-4)*3=36 Ans. 36
➢Type 10 - Factorial Based Series
This is the latest pattern question asked in latest exams.
Example 1: Ans. 33
Example 2: Ans. 606 + 721 = 1327
Question 8:In the given series, only one number is wrong. 3, 9, 19, 32 Find out the wrong number.
Explanation
- As we know, the most crucial step towards solving such type of questions is to guess the rule with which the series has been put together. The first two terms don’t seem to be of much help. Remember either of these could be wrong, so to guess the rule of the series you have to always consider more than two terms (in case there are four). - In other words, more than half the number of terms that have been given must satisfy the rule that you come up with. - If you check carefully, you will see that the rule in the above example is (2n2 + 1) where n starts from 1 i.e. the term number. Hence the term that doesn’t comply with this rule is the last term which should be equal to 33 and not 32. therefore, the correct answer is D) 32.
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