In mathematical terminology, the terms “sequence” and “series” have slightly different meaning. In simple words, a sequence is a comma-separated list of numbers or objects following a special order, whereas, a series is the sum of terms of a sequence. However, here the term “series formation” implies formation of both sequence and series.

In the questions on this section, usually a sequence of letters, numbers or symbols following some particular predefined rules will be given in the question. By understanding and applying these predefined rules, you are expected to find out the succeeding terms or a missing term in the given sequence.

Sequence can be categorized as:

- Number Sequence
- Letter Sequence
- Symbol Sequence

# 1 Number Sequence

Some of the important types of number sequence are:

## Arithmetic sequence

Here, successive numbers are obtained by adding (or subtracting) a fixed number to the previous number. The fixed number is called the common difference of the arithmetic sequence.

**Examples: **

- 2, 4, 6, 8,….
- 1, 6, 11, 16,….
- 10, 8, 6, 4,…

## Geometric Sequence

A geometric sequence is the one in which each successive number is obtained by multiplying (or dividing) a fixed number (can be fraction also) by the previous number. That fixed number is called the common ratio of the geometric sequence.

**Examples:**

- 3, 6, 12, 24,…
- 2048, 1024, 512, 256, 128, 64,…
- 10, -40, 160, -640,…
- 1024, 640, 400, 250,…

*Trick:** A simple trick to identify a geometric sequence is by dividing each pair of successive numbers in that sequence and checking whether we are getting a common ratio.*

## Prime Sequence

As we know, prime numbers are numbers which can be exactly divided only by one and that number. Prime sequence is the ordered set of prime numbers starting with any of the prime numbers.

**Examples:**

- 7, 11, 13, 17, 19,…
- 29, 31, 37, 41, 43,…

## Power Sequence

In this sequence, every term will be the n^{th} power of some consecutive numbers. It can be square, cube or any higher powers of a particular sequence of numbers.

**Examples: **

- 1, 4, 9, 16, 25… – Square sequence
- 1, 8, 27, 64, 125, … – Cube sequence
- 1, 16, 81, 256… – 4
^{th}power sequence.

## Reversal Sequence

In this sequence, reversing each term will give us a sequence, which can be any of the above-mentioned sequence.

**Example:**

- 31, 51, 71, 91, 12…
- 61, 52, 63, 94, 46, 18,..

## Two-tier sequence

In a two-tier sequence, the difference of the successive terms can form any of the above-mentioned sequence like arithmetic sequence, prime sequence, power sequence etc.

## Twin sequence

Twin sequence are the one in which two sequence are packed into one.

In a twin sequence any of the above-mentioned sequence like arithmetic sequence, prime sequence, etc. can occur.

**Examples:**

- 1, 0, 3, 2, 5, 4, 7,…
- 3, 1, 6, 3, 12, 5, 24,…

# 2 Letter Sequence

In this type of sequence, a number of letters are arranged in a sequence and we are supposed to find the next letter or letters in the sequence.

The easiest way to solve these questions is by converting the letters into their numerical equivalent. That is, A = 1, B = 2, C = 3, … , X = 24, Y = 25, Z = 26.

After converting to numerical equivalent, the sequence becomes same as a number sequence. Finding the next number in a ‘Number Sequence’ is already discussed above.

*Note that while finding the next number, if there occur a number, say x, greater than **26**, subtract **26** from x to get the required number. In addition, if there occur a negative number, add **26** to it to get the required number.*

*Trick**:* For converting an alphabet to the corresponding number or vice-versa, it is convenient to remember one or two equivalents. It saves lot of time.

Say, **remember that J ≡ ****10**** and T ≡ ****20****.**

Now any letter can easily be compared to these two letters.

For example, to find the numerical equivalent of V:

V is the 2^{nd} number from T.

Therefore, V is equivalent to 20 + 2 = 22.

# 3 Symbol Sequence

In a symbol sequence, a pattern is given for three or four figures; we have to decide which will be the next figure to complete the given sequence.

**Tips and tricks for identifying the sequence quickly**

- If the sequence is increasing or decreasing with a small increase there must be an addition or subtraction involved, possibly an arithmetic sequence.

** Example:** 1, 3, 5, 7,…

- If the sequence is increasing or decreasing with a large increase there must be a multiplication or division involved, possibly a geometric sequence.

** Example:** 4, 8, 16, 32,…

- If the sequence is increasing or decreasing with a very large increase there must be square or cubes involved, possibly a power sequence.

** Example:** 36, 49, 64, 81,…

- If the sequence is not linearly increasing and a pattern is not formed after solving for three terms then there is likely a combination of sequence, That is, a twin sequence.

** Example:** 1, 0, 3, 2, 5, 4, 7…

- If a pattern is not formed after solving for 3 terms, then the pattern may involve an operation of first and second terms to give the third term. Even it may involve grouping of terms.

** Example:** 2, 3, 5, 6, 3, 4, 7, 12, 4, 5, 9…

Take groups of four. That is, (2,3,5,6), (3,4,7,12), (4,5,9,…)

- If second term cannot be formed from first term or third term cannot be formed from first and second terms, then likely an operation is done on another sequence to get this sequence.

** Example:** 1, 4, 9, 16,… is a square of 1, 2, 3, 4,…

- For symbol sequence, by observation, eliminate improbable options to save time.

*“For detailed theory, refer the book “CSIR-NET General Aptitude – A New Outlook”*