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# Number Series Formula

**Formulas for Number series**

Here on this page you will get to know about the Number Series formulas with its definition and types. You will also get to learn, How to solve Questions with the help of formulas.

**Number Series Formulas:**

Number series is a sequence of numbers which is framed, using a particular system/rule, in which the numbers are not in a specific order. All we have to do is to find out the system/rule that a particular series is following and finding the number using that system/rule.

**For Example :** 3, 9, 27, 81 ?

It is Geometric series.

Each term is Multiplied by 3.

So , 81 x 3 = 243

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**Formulas for Number Series :**

Arithmetic Progression | Geometric Progression | |
---|---|---|

Sequence | a, a+d, a+2d,……,a+(n-1)d,…. | a, ar, ar^{2},….,ar^{(n-1)},… |

Common Difference or Ratio | Successive term – Preceding term Common difference = d = a2 – a1 | Successive term/Preceding term Common ratio = r = ar^{(n-1)}/ar^{(n-2)} |

General Term (nth Term) | an = a + (n-1)d | an = ar^{(n-1)} |

nth term from the last term | an = l – (n-1)d | an = l/r^{(n-1)} |

Sum of first n terms | sn = n/2(2a + (n-1)d) | sn = a(1 – rn)/(1 – r) if |r| < 1 sn = a(rn -1)/(r – 1) if |r| > 1 |

### Types of number series:-

Types of Number Series is given below:

**Arithmetic Sequence :**A sequence in which every term is created by adding or subtracting a definite number to the preceding number is an arithmetic sequence.**Geometric Sequence :**A sequence in which every term is obtained by multiplying or dividing a definite number with the preceding number is known as a geometric sequence.**Harmonic Sequences****:**

A series of numbers is said to be in harmonic sequence if the reciprocals of all the elements of the sequence form an arithmetic sequence.**Fibonacci Numbers :**Fibonacci numbers form an interesting sequence of numbers in which each element is obtained by adding two preceding elements and the sequence starts with 0 and 1.

**Add up Series +**

- Probability of asking – Very Low
- Difficulty – Low
- Reason – to Introduce Concept

These problems are never asked they are very easy, we are talking about them to introduce the number series from basic.

- Rule – Just Add a number ‘N’ to the last number.
- E.g. – 5, (5 + 3 = 8), (8 + 3 = 11), ( 11 + 3= 14) ….
- Result – 5, 8, 11, 14, 17 ……..

**Add up Series –**

- Probability of asking – Very Low
- Difficulty – Low
- Reason – to Introduce Concept

- Rule : Just Add a number ‘N’ to the last number.
- E.g. : 4, (4 – 5 = (-1)), (-1 -5 = -6), ( -6 – 5 = -11) ….
- Result : 4, -1, -6, -11, -16 ……..

**Square up +(Easy to Identify)**

- Rule – For a number X and for a number a where a = 1, 2, 3….. do next number = x + a
^{2} - E.g. – 5
- 5 + 2
^{2 }= 5 + 4 = 9 - 9 + 3
^{2}= 9 + 9 = 18 - 18 + 4
^{2}= 18 + 16 = 34 - 34 + 5
^{2}= 34 + 25 = 59

- 5 + 2
- Result – 5, 9, 18, 34, 59 …..

**Square up Add up +(Hard to Identify)**

- Rule – For a number X and for a number a where a = 1, 2, 3….. do next number = x + a
^{2 }+ b for b some pattern. - E.g. – 5
- 5 + 2
^{2 }+ 3 = 5 + 4 + 3 = 12 - 12 + 3
^{2}+ 3 = 12 + 9 + 3 = 24 - 24 + 4
^{2}+ 3 = 24 + 16 + 3 = 43 - 43 + 5
^{2}+ 3 = 43 + 25 + 3 = 71

- 5 + 2
- Result – 5, 12, 24, 43, 71 …..

**Square up Step up +(Very hard to identify not asked mostly unless paper is very tough)**

- Rule – For a number X and for a number a where a = 1, 2, 3….. do next number = x + a^{2} + b for b some pattern.
- E.g. – 5
- 5 + 2^{2} + 3 = 5 + 4 + 3 = 12
- 12 + 3^{2} + 8(3+5) = 12 + 9 + 8 = 29
- 29 + 4^{2} +13(8+5) = 29 + 16 + 13 = 58
- 58 + 5^{2} + 18(13+5) = 58 + 25 + 18 = 101

- Result – 5, 12, 29, 58, 101 ..

Same for Step Up Series +, but instead of adding, Subtract.

**Questions and Answers for Number Series :**

**Question : **Find the missing number? 99, 121, 143, ___, 187, 199 .

A. 170

B. 165

C. 158

D. 172

**Answer : 165**

Explanation:

The given series is an AP with first term as 99 and common difference as 22.

**Question : **Find the next term in the series : 51,52,53,55,58,63,____.

A. 69

B. 77

C. 81

D. 71

**Answer : 71**

Explanation:

Fibonacci series is added to each term.

51 + 0 =51

51 + 1=52

52 + 1=53

53 + 2=55

55 + 3=58

58 + 5=63

63 + 8=71

**Question : **Find the missing terms? 97,122,107,132,__,__.

A. 117,142

B. 122,112

C. 141,131

D. 121,131

**Answer : 117,142**

Explanation:

This series is a result of alternate +25 and -15.

97 + 25=122

122 – 15=107

107 + 25=132

132 – 15=117

117 + 25=142

So, the next two terms are 117 and 142.

**Question : **Fill the missing term in the series

100, 92, 86 ,82, 74, 68, 64, 56, 50, __, ___.

A. 44, 36

B. 40, 34

C. 46, 38

D. 44, 32

**Answer : 46, 38**

Explanation:

The number series are in successive subtraction series of – 8, -6, -4 and then again -8,-6,-4.

So, the next terms after 50 will be 50-4=46 and 46-8=38.

**Question : **Select the missing number from the given responses.

19, 35, 67, 131, 259, 515, ?

A. 1281

B. 1291

C. 1071

D. 1027

Explanation :

11 × 2 – 3 = 19

19 × 2 – 3 = 35

35 × 2 – 3 = 67

67 × 2 – 3 = 131

131 × 2 – 3 = 259

259 × 2 – 3 = 515

515× 2 – 3 = 1027

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- Coding and Number Series – Questions | Formulas | How to Solve Quickly | Tricks & Shortcuts
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