PRIME NUMBERS : A prime number is a a counting number greater then 1, which is divisible only by 1 and by itself. For example: 2, 3, 5, 7, 11, etc.

PERFECT SQUARE : A whole number whose square root is also a whole number is called a perfect square. For example: 4, 9, 16, etc.

PERFECT CUBE : A whole number whose cube root is also a whole number is called a perfect cube. For example: 1, 8, 27, 64, etc.

Arithmetic Progression (AP)/ Arithmetic Series

An arithmetic progression is the sequence of numbers in which a number in the series is obtained by adding a constant number to the previous number in the series.
The sum of first n terms of the series s=n2{2a+(n−1)d}
Where a is the first term and d is the common difference.
The sum of the series (of all n terms)
= n2{a+l}=n2{2a+(n−1)d}
Where ‘a’ is the first term and ‘l’ is the last term

For example,
2, 5, 8, 11 ……. n common difference = 3
9, 15, 21, 27………. n common difference = 6

Geometric Progression (GP) / Geometric Series

A sequence of numbers is said to be in geometric progression (GP), if the ratio of two consecutive terms in the sequence is always the same and is called the common ratio. The nth term of a series is given by
an=a. rn−1
l=a. rN−1
Sn=a(rn−1)/(r−1)
SN=a(rN−1)/(r−1),r≠1

Where,
a = first term sn = Sum of first n terms
r = common ratio sN = Sum of all N terms of the series
l = last term

For examples
3, 15, 75, 375, ………. n common ratio = 5
2,6, 18, 54 ………. common ratio = 3

Combination Series

This can be series combing different rules in a logical way.

(a) Series of perfect squares: For example 9, 25, 49, 81, 121………
The series can be represented by the mathematical expression f(x)=(1+2x)2
Where x = 1, 2, 3

(b) Series of perfect cubes : For example,
(i) 1, 27, 729, 19683……..
Can be represented by f(x)=(27)x−1
Where x=1,2,3,4

(ii) 1, 64, 373, 1000…
Can be represented by fx=[1+3(x−1)]3
Where x = 1, 2, 3, 4…..

Second Degree Arithmetic Series

A series in which the differences between the successive terms themselves are in AP and forms an arithmetic series is called the second degree arithmetic series.

For example, 2, 3, 6, 11, 18……

A series in which the differences between the successive terms themselves are in AP and forms an arithmetic series is called the second degree arithmetic series.

For example, 2, 3, 6, 11, 18……

The Difference of successive terms of the above series 1, 3, 5, 7..... which forms an arithmetic series with common difference 2.
The given series can be represented by a quadratic mathematics function: f(x)=x2+2
Where x =0, 1, 2, 3, 4…….

Third Degree Arithmetic Series

A series in which the difference of the successive terms forms a second degree arithmetic series is termed third degree arithmetic series. Mathematically, such a series can be represented by a general cubic function and hence the name third degree.

For example:
(a) 20, 30, 42, 59, 84, 120, 170,…
The difference of successive terms of the above series is 2, 5, 11, 14… and this forms an arithmetic series with common difference 3.

(b) 1, 15, 53, 127,249, 431, 685,……
The difference of successive terms of the above series is 14, 38, 74, 122, 182, 254. The difference of successive terms of the above series is 24, 36, 48, 60, 72 and this forms an arithmetic series with common difference 12.

Mathematically, the given series can be represented by a cubic function
f(x)=2x3−2
Where x = 1, 2, 3, 4…..

Arithmetic-Geometric (A-G) Series

The series is formed by combination of arithmetic progressions. Each succeeding term is obtained by first adding ( or subtracting) the common difference of its AP to the previous term and then multiplying (or dividing) the sum by common ratio of its GP. An important property of such an A-G series is that the difference of successive terms is in GP.

For example: 1, 8, 22, 50, 106, 218
Series is obtained by adding 3 to previous term and then multiplying the sum by 2.
i.e a2=(a1+3)×2

Geometric Arithmetic (G-A) Series

Each succeeding term in G-A series is funded by first multiplying(or dividing)
the previous term by the common ratio of its GP and then adding (or subtracting) to it the common difference of its AP. This important property of G-A series, like A-G series, is that the difference of its successive terms are in geometric progression.

For example: 1, 2, 6, 22, 86, 342,..
The terms in the above series are obtained by fist multiplying the previous term by 4 and then subtracting 2 from the resultant product.

Twin Series

As the name suggests, it consists of two series combined into a single series. The alternating terms of this series form an independent series.

For example: 2, 3, 7, 9, 12, 27, 17, 81, 22, 243…..
Odd terms of the series 2, 7, 12, 22,…… is an arithmetic series with common difference 5. Even terms of the series 3, 9, 27, 81, 243….. is a geometric series with common ratio 3.

Alternating Series

In an alternating series, the successive terms increase and decrease alternately. It differs from irregular increase (or decrease) series wherein the successive terms increase (or decrease) continuously but the ratio of increase does not follow a particular trend. While solving an alternate series we should focus on the following possibility of its formation:
1) Whether it is a twin series which has been formed by combining two different series.
2) It has been formed by performing two different kinds of operations alternately on successive terms.

For example: 6, 13, 12, 40, 24, 104, 48, 229, 96…

In the above series, there are two series 6, 12, 24, 48, 96…and 13, 40, 104,229...i.e. the fist one is a geometric series with first term 6 and common ratio 2.
In the second series, terms are having difference 27, 64, 125…. i.e…, 33,45,53

Odd Man Out (Wrong Number) in a Series

For example:

3, 5, 7, 9, 11…. 9 is the odd man out because all others are prime numbers.
2, 3, 7, 10, 11… Except 10, all are prime numbers.
3, 10, 29, 66, 127, 220, 345….. Except 220, all other terms can be expressed as x3+2, where x = 1, 2, 3…..

Special Type of Series

(a) Numbers followed by their LCM or HCF.

For example: 1, 2, 3, 6, 4, 5, 12, 60, 5, 6, 7 ….
In the above series, every four numbers forms a block in which the fourth number is the LCM of first three numbers. Therefore, the number next to 7 is 210. i.e. LCM of 5, 6, 7 = 210

(b) Numbers followed by their product.

For example:
1, 2, 2, 4, 8, 32, ……
1×2=2;2×2=4;2×4=8;4×8=32;8×32=256,
Which is the next number in the series.

(c) By use of digit sum. For example: 12, 15, 21, 24, 30, ..
Next number = previous number + Digit-sum of previous number.
i.e., 15 = 12+(1+2)
21=15+(1+5)
24=21+(2+1)
30=24+(2+4)
∴ Next number = 30 +(3+0)= 33

Number Series Questions from Previous Year Exams

Number Series Video Lecture.

https://www.youtube.com/watch?v=pHm1qepw5Ro