Introduction
Number Series is one of the most important topics in reasoning for competitive exams. In number series questions, you need to identify the hidden pattern between numbers and find the next number, missing number, or wrong term in the series. These patterns may be based on addition, subtraction, multiplication, division, squares, cubes, Fibonacci series, step differences, or mixed operations. With regular practice, students can solve number-series questions quickly and accurately.
Number series questions are commonly asked in competitive exams like SSC, Banking, Railways, UPSC, Police, Defence, Insurance, and many other government exams. These questions are considered scoring because they can often be solved in less time using proper methods and shortcuts. In this article, we will learn some important Number Series Tricks and concepts that help you understand patterns easily and solve reasoning questions faster in exams.
Number Series Tricks (Basic Series)
Addition
1. Logic
Numbers increase by adding a fixed number or changing additions.
2. Example
2, 5, 8, 11, 14, ?
Pattern: +3 each time
2 + 3 = 5
5 + 3 = 8
8 + 3 = 11
11 + 3 = 14
14 + 3 = 17
Answer = 17
3. Another Example (changing addition)
3, 7, 12, 18, 25, ?
Pattern: +4, +5, +6, +7
Next = +8
25 + 8 = 33
Substraction
1. Logic
Numbers decrease by subtracting a fixed number or changing numbers.
2. Example (fixed subtraction)
25, 22, 19, 16, 13, ?
Pattern: −3 each time
25 − 3 = 22
22 − 3 = 19
19 − 3 = 16
16 − 3 = 13
13 − 3 = 10
Answer = 10
3. Example (changing subtraction)
50, 45, 39, 32, 24, ?
Pattern: −5, −6, −7, −8
Next = −9
24 − 9 = 15
Multiplication
1. Logic
Numbers increase or decrease by multiplying with a number.
2. Example (fixed multiplication)
2, 4, 8, 16, 32, ?
Pattern: ×2 each time
2 × 2 = 4
4 × 2 = 8
8 × 2 = 16
16 × 2 = 32
32 × 2 = 64
Answer = 64
3. Example (changing multiplication)
2, 6, 24, 120, ?
Pattern:
2 × 3 = 6
6 × 4 = 24
24 × 5 = 120
120 × 6 = 720
Answer = 720
Division
1. Logic
Numbers decrease by dividing with a number.
2. Example (fixed division)
128, 64, 32, 16, 8, ?
Pattern: ÷2 each time
128 ÷ 2 = 64
64 ÷ 2 = 32
32 ÷ 2 = 16
16 ÷ 2 = 8
8 ÷ 2 = 4
Answer = 4
3. Example (changing division)
720, 120, 24, 6, 2, ?
Pattern:
720 ÷ 6 = 120
120 ÷ 5 = 24
24 ÷ 4 = 6
6 ÷ 3 = 2
2 ÷ 2 = 1
Answer = 1
Squares
1. Logic
Numbers follow square values:
1², 2², 3², 4², 5² …
Square numbers:
1, 4, 9, 16, 25, 36, 49, 64, 81, 100 …
2. Example
1, 4, 9, 16, 25, ?
Pattern: square numbers
1² = 1
2² = 4
3² = 9
4² = 16
5² = 25
6² = 36
Answer = 36
3. Example (mixed pattern)
5, 8, 13, 20, 29, ?
Differences:
+3, +5, +7, +9
Next = +11
29 + 11 = 40
(odd numbers are added continuously, related to square patterns)
4. What to check first in exams
- Perfect squares?
- Differences as odd numbers?
- Squares + addition/subtraction?
- Alternate square patterns?
5. Important Squares to Memorize
1² = 1
2² = 4
3² = 9
4² = 16
5² = 25
10² = 100
15² = 225
20² = 400
25² = 625
30² = 900
6. Exam Trick
If numbers increase in a special pattern:
- Check square numbers first
- Then check differences between terms
- Odd-number additions often indicate squares
This method makes square-based number series easy for competitive exams.
Cubes
1. Logic
Numbers follow cube values:
1³, 2³, 3³, 4³, 5³ …
Cube numbers:
1, 8, 27, 64, 125, 216, 343, 512, 729, 1000 …
2. Example
1, 8, 27, 64, 125, ?
Pattern: cube numbers
1³ = 1
2³ = 8
3³ = 27
4³ = 64
5³ = 125
6³ = 216
Answer = 216
3. Example (mixed cube pattern)
9, 16, 35, 72, 133, ?
Differences:
+7, +19, +37, +61
These are related to cube growth patterns.
4. What to check first in exams
- Perfect cubes?
- Cubes + addition/subtraction?
- Alternate cube patterns?
- Differences increasing rapidly?
5. Important Cubes to Memorize
1³ = 1
2³ = 8
3³ = 27
4³ = 64
5³ = 125
6³ = 216
7³ = 343
8³ = 512
9³ = 729
10³ = 1000
6. Exam Trick
If numbers increase very fast:
- Check cube numbers first
- Then factorial or multiplication patterns
- Cubes usually grow much faster than squares
This method helps you solve cube-based number series quickly in competitive exams.
Number Series Tricks (Advance series)
Multiplication + Addition logic
1. Logic
Each number is formed by:
Previous number × something + something
2. Example (fixed pattern)
2, 5, 11, 23, 47, ?
Pattern:
2 × 2 + 1 = 5
5 × 2 + 1 = 11
11 × 2 + 1 = 23
23 × 2 + 1 = 47
47 × 2 + 1 = 95
Answer = 95
3. Example (changing addition)
3, 7, 15, 31, 63, ?
Pattern:
3 × 2 + 1 = 7
7 × 2 + 1 = 15
15 × 2 + 1 = 31
31 × 2 + 1 = 63
63 × 2 + 1 = 127
Answer = 127
4. Another Example
2, 6, 15, 31, 56, ?
Pattern:
2 × 2 + 2 = 6
6 × 2 + 3 = 15
15 × 2 + 1 = 31
(look carefully — sometimes mixed patterns appear)
5. What to check first in exams
- ×2 +1 pattern?
- ×3 +2 pattern?
- Increasing additions after multiplication?
- Alternate multiplication and addition?
6. Exam Trick
If numbers increase very fast but not exact multiplication:
- Check multiplication first
- Then see if a small number is added/subtracted
Example:
8 → 17
8 × 2 = 16
16 + 1 = 17
This method makes mixed-operation number series easy for competitive exams.
Multiplication + Subtraction logic
1. Logic
Each number is formed by:
Previous number × something − something
2. Example (fixed pattern)
5, 9, 17, 33, 65, ?
Pattern:
5 × 2 − 1 = 9
9 × 2 − 1 = 17
17 × 2 − 1 = 33
33 × 2 − 1 = 65
65 × 2 − 1 = 129
Answer = 129
3. Example (changing subtraction)
3, 8, 22, 63, 185, ?
Pattern:
3 × 3 − 1 = 8
8 × 3 − 2 = 22
22 × 3 − 3 = 63
63 × 3 − 4 = 185
185 × 3 − 5 = 550
Answer = 550
4. What to check first in exams
- ×2 −1 pattern?
- ×3 −2 pattern?
- Increasing subtraction after multiplication?
- Alternate multiplication and subtraction?
5. Exam Trick
If numbers increase quickly but are slightly smaller than multiplication results:
- Check multiplication first
- Then see if a small number is subtracted
Example:
20 → 39
20 × 2 = 40
40 − 1 = 39
This method helps you solve multiplication + subtraction series quickly in competitive exams.
Step Difference
Step Difference in number series means finding the difference between consecutive numbers to identify the hidden pattern. It is one of the most important and easiest methods used in reasoning for competitive exams like SSC, Banking, Railways, UPSC, and others.
In this method, you subtract one term from the next term:
Difference = Next number − Previous number
After finding the differences, you observe whether the differences follow a pattern such as:
- increasing numbers,
- decreasing numbers,
- squares,
- cubes,
- prime numbers,
- odd/even numbers,
- multiplication patterns, etc.
Example 1
2, 5, 9, 14, 20, ?
Find differences:
5 − 2 = 3
9 − 5 = 4
14 − 9 = 5
20 − 14 = 6
Pattern: +3, +4, +5, +6
Next difference = +7
20 + 7 = 27
Answer = 27
Here, the original series does not directly show the pattern. The hidden logic appears only after finding the step differences.
Example 2
3, 7, 13, 21, 31, ?
Differences:
7 − 3 = 4
13 − 7 = 6
21 − 13 = 8
31 − 21 = 10
Pattern: even numbers increasing by 2
Next difference = 12
31 + 12 = 43
Answer = 43
Why Step Difference is Important
Many number series questions are not simple addition or multiplication. Examiners hide the pattern inside the differences. So, whenever you cannot identify the pattern directly, the first thing you should do is check the differences between numbers.
What to Check in Differences
After finding differences, check whether they are:
- consecutive numbers (2,3,4,5)
- odd numbers (3,5,7,9)
- even numbers (2,4,6,8)
- squares (1,4,9,16)
- cubes (1,8,27)
- prime numbers (2,3,5,7)
- multiplication patterns
Exam Trick
If numbers are increasing slowly or irregularly:
- First check addition/subtraction
- Then immediately find step differences
- If needed, find second differences also
Example of Second Difference
1, 4, 9, 16, 25
Differences:
3, 5, 7, 9
Again differences:
2, 2, 2
This confirms a square-number pattern.
Final Tip
In reasoning exams, around half of number-series questions become easy once you learn step differences properly. Practice finding differences quickly, and always observe patterns carefully before choosing the answer.
N² ± A Logic
n² ± a different number pattern is a very common number-series logic in competitive exams. In this type, the series is based on square numbers, but an extra number is either added or subtracted.
Formula style:
- n² + number
- n² − number
Here, n means natural numbers:
1², 2², 3², 4², 5² …
Square numbers:
1, 4, 9, 16, 25, 36, 49 …
Then another number is added or subtracted.
Example 1: n² + number
2, 6, 12, 20, 30, ?
Check squares:
1² = 1 → +1 = 2
2² = 4 → +2 = 6
3² = 9 → +3 = 12
4² = 16 → +4 = 20
5² = 25 → +5 = 30
Next:
6² = 36
36 + 6 = 42
Answer = 42
Example 2: n² − number
0, 2, 6, 12, 20, ?
Check pattern:
1² − 1 = 0
2² − 2 = 2
3² − 3 = 6
4² − 4 = 12
5² − 5 = 20
Next:
6² − 6 = 30
Answer = 30
Example 3: Different added numbers
3, 7, 14, 24, 37, ?
Squares:
1² + 2 = 3
2² + 3 = 7
3² + 5 = 14
4² + 8 = 24
5² + 12 = 37
Here, added numbers also follow a pattern.
How to Identify This Pattern in Exams
When numbers:
- increase moderately,
- are not simple addition/multiplication,
- and look close to square numbers,
then check:
- nearest perfect square,
- what number is added/subtracted.
Example:
26 is close to 25
50 is close to 49
65 is close to 64
This often indicates square logic.
Important Squares to Remember
1² = 1
2² = 4
3² = 9
4² = 16
5² = 25
6² = 36
7² = 49
8² = 64
9² = 81
10² = 100
Exam Trick
If differences increase regularly:
- first check square numbers,
- then see whether a small number is added/subtracted.
This pattern is very common in SSC, Banking, Railways, and other reasoning exams.
N² ± N Logic
n² ± n pattern is a very important number-series concept in reasoning. In this pattern, each term is formed using:
- n² + n
or - n² − n
Here, n means natural numbers:
1, 2, 3, 4, 5 …
And square numbers are:
1², 2², 3², 4², 5² …
1. n² + n Pattern
Formula:
n^2+n
Example
2, 6, 12, 20, 30, ?
Check pattern:
1² + 1 = 2
2² + 2 = 6
3² + 3 = 12
4² + 4 = 20
5² + 5 = 30
Next:
6² + 6 = 42
Answer = 42
2. n² − n Pattern
Formula:
n^2-n
Example
0, 2, 6, 12, 20, ?
Check pattern:
1² − 1 = 0
2² − 2 = 2
3² − 3 = 6
4² − 4 = 12
5² − 5 = 20
Next:
6² − 6 = 30
Answer = 30
How to Identify Quickly in Exams
Check whether numbers are close to square numbers:
- 20 is close to 16 or 25
- 30 is close to 25 or 36
- 42 is close to 36
Then test:
- square + n
- square − n
Step Difference Trick
For n² + n:
2, 6, 12, 20, 30
Differences:
4, 6, 8, 10
Even numbers increase regularly.
For n² − n:
0, 2, 6, 12, 20
Differences:
2, 4, 6, 8
Again even-number pattern appears.
Exam Tip
Whenever:
- differences increase regularly,
- numbers look near square values,
- and simple addition/multiplication fails,
check:
- n² + n
- n² − n
This pattern is very common in SSC, Banking, Railways, and other competitive exams.
N³ ± A Logic
n³ ± a pattern is an important number-series logic in reasoning. In this type, the series is based on cube numbers, and a fixed or changing number is added or subtracted.
Cube numbers are:
1³ = 1
2³ = 8
3³ = 27
4³ = 64
5³ = 125
6³ = 216
Formula style:
- n³ + a
- n³ − a
where:
- n = natural number
- a = fixed or changing number
1. n³ + a Pattern
Formula:
n^3+a
Example
2, 9, 28, 65, 126, ?
Check pattern:
1³ + 1 = 2
2³ + 1 = 9
3³ + 1 = 28
4³ + 1 = 65
5³ + 1 = 126
Next:
6³ + 1 = 217
Answer = 217
2. n³ − a Pattern
Formula:
n^3-a
Example
0, 7, 26, 63, 124, ?
Check pattern:
1³ − 1 = 0
2³ − 1 = 7
3³ − 1 = 26
4³ − 1 = 63
5³ − 1 = 124
Next:
6³ − 1 = 215
Answer = 215
3. n³ ± changing number
Example
3, 10, 29, 68, ?
Check:
1³ + 2 = 3
2³ + 2 = 10
3³ + 2 = 29
4³ + 4 = 68
Sometimes added/subtracted numbers also follow another pattern.
How to Identify Quickly in Exams
If numbers:
- increase very fast,
- are close to cube values,
- and multiplication patterns fail,
then check cube logic.
Example:
28 is near 27
65 is near 64
126 is near 125
This indicates cube-based series.
Step Difference Trick
Example:
2, 9, 28, 65, 126
Differences:
7, 19, 37, 61
Differences increase rapidly — a common sign of cube logic.
Exam Tip
Whenever numbers grow very quickly:
- Check multiplication
- Then check cubes
- Then see if a small number is added/subtracted
Cube-based patterns are very common in SSC, Banking, Railways, and other competitive exams.
N³ ± N
n³ ± n pattern is a very important number-series logic in reasoning. In this type, each term is formed using:
- n³ + n
or - n³ − n
where:
- n = natural numbers (1, 2, 3, 4…)
- cube numbers are:
1³ = 1
2³ = 8
3³ = 27
4³ = 64
5³ = 125
1. n³ + n Pattern
Formula:
n^3+n
Example
2, 10, 30, 68, 130, ?
Check pattern:
1³ + 1 = 2
2³ + 2 = 10
3³ + 3 = 30
4³ + 4 = 68
5³ + 5 = 130
Next:
6³ + 6 = 222
Answer = 222
2. n³ − n Pattern
Formula:
n^3-n
Example
0, 6, 24, 60, 120, ?
Check pattern:
1³ − 1 = 0
2³ − 2 = 6
3³ − 3 = 24
4³ − 4 = 60
5³ − 5 = 120
Next:
6³ − 6 = 210
Answer = 210
How to Identify Quickly in Exams
If numbers:
- increase very fast,
- are close to cube numbers,
- and simple multiplication does not match,
then check:
- cube + n
- cube − n
Example:
68 is close to 64
130 is close to 125
222 is close to 216
Step Difference Trick
Example:
2, 10, 30, 68, 130
Differences:
8, 20, 38, 62
Differences increase rapidly — a common sign of cube-based logic.
Important Cubes to Remember
1³ = 1
2³ = 8
3³ = 27
4³ = 64
5³ = 125
6³ = 216
7³ = 343
8³ = 512
9³ = 729
10³ = 1000
Exam Tip
Whenever numbers increase very quickly:
- Check multiplication
- Then check cube numbers
- Then see whether n is added/subtracted
This pattern is commonly asked in SSC, Banking, Railways, and other competitive exams.
