Number series are mathematical sequences that follow a certain rule based on elementary arithmetic operations. A series or sequence of numbers is given, with 1 or more missing terms. One must find the pattern (or find the logic) that connects the numbers forming the series, apply the logic or pattern to find out the missing number(s). Some of the commonly asked patterns are listed below-

__Series with__

__Same Difference__

**3, 6, 9, 12, _** (15 is the next term obtained by adding 3 to previous term)

**Ans 15**

**24, 20, 16, 12, _** (8 is the next term obtained by reducing 4 from the previous term)

**Ans 4**

__Increasing Difference__

**3, 4, 6, 9, __** (The difference between successive numbers in increasing by 1, 2, 3,… So, the missing term 13 obtained by adding 4 to the previous term 9)

**Ans 13**

**25, 24, 22, 19, _** (The difference between successive numbers is increasing by 1, 2, 3…. So, the missing term 15 obtained by reducing 4 from the previous term 19)

**Ans 15**

__Decreasing Difference __

**2, 9, 15, 20, __** (The difference between successive numbers in decreasing by 7, 6, 5… so, the missing term 24 obtained by adding 4 to the previous term 20)

**Ans 24**

**30, 22, 15, 9, __** (The difference between successive numbers is decreasing by 8, 7, 6…. so, the missing term 4 obtained by reducing 5 from the previous term 9)

**Ans 4**

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__Difference Doubles or Halves__

**5, 9, 17, 33, __.** (The difference between every two successive numbers is double of the difference between previous pair of successive terms 9-5=4, 17-9=8, 33-17=16…., so, the missing term 65 is obtained by adding 32 to the previous term 33)

**Ans:65**

**120, 80, 60, 50, _ **(The difference between successive numbers is half of the previous difference i.e.120-80= 40, 80-60=20, 60-10=50.., so the missing term 45 is obtained by reducing 5 from the previous term 50)

**Ans: 45**

__Multiplication or Division By Same Number __

**3, 6, 12, 24, _** (The successive numbers obtained by multiplying the previous number by 2, so the missing term 48 is obtained by multiplying 24 by 2)

**Ans: 48**

**729, 243, 81, 27, _ **(The successive numbers are obtained by dividing the previous number by 3, so the missing term 9 is obtained by dividing 27 by 3)

**Ans: 9**

__Multiplication and Division is Done By Increasing Numbers__

**2, 4, 12, 48, _** (The successive numbers are obtained by multiplying the previous numbers by 2,3,4,…. so the missing term 240 is obtained by multiplying 48 by 5)

**Ans: 240**

**120, 60, 20, 5, _** (The successive numbers are obtained by dividing the previous numbers

by 2,3,4,…, so the missing term 1 is obtained by dividing 5 by 5)

**Ans: 1**

__Previous Two Numbers Add Up to Third Number __

**1,1, 2, 3, 5, _** (The successive numbers are obtained by adding the previous two numbers, so, the missing term 8 is obtained by adding 3 and 5)

This is also known as **Fibonacci series**.

**Ans: 8**

__Successive Number is Product of Previous Two Numbers __

**1, 3, 3, 9, 27,_** (The successive numbers are obtained by multiplying the previous two numbers, so the missing term 243 obtained by product of 9 and 27)

**Ans: 243**

__Mixed Series or Series formed by combination of two series __

The missing term belongs to series 2: **Ans- 32 ** is obtained by multiplying 16 by 2

__Combination of mathematical operations __

**2, 3, 8, 27, __ **

The missing term 112 is obtained by following the mathematical operation (27 X 4) + 4

**Ans: 112**

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__Miscellaneous Patterns__

Number series and patterns can come in an unimaginable variety. Lets try a few more patterns

Some examples:

**Question** : Write the missing term from the given options.

- 2, 9, 30, 68, ___, 222.

a)112 b) 124 c) 130 d) 180

**Solution:** 1^{3}+1=2 ,^{ }2^{3} + 2=10 , 3^{3}+3=30 , 4^{3}+4=68 , 5^{3}+5=130 , 6^{3}+6=222

2. 165, 275, 385, 495, ___ , 660

a)585 b) 561 c) 530 d) 541

**Solution:** Each number of the series has digit at 1^{s} place +digit at 100^{s} place= digit at 10^{s} place. Only option b) fits the pattern 5+1=6.

Ans **b) 561**

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3. 143,77, 35, 15, ___

a)5 b) 8 c) 6 d) 0

**Solution:** 13×11=143, 11×7=77, 7×5=35, 5×3=15, 3×2=6 . Product of two successive prime numbers taken in descending order.

Ans: **c) 6**