How to Solve Number Series Using Pattern Methods?
Number Series consists of numbers arranged according to a definite pattern or rule. The missing or incorrect term is found by examining differences, ratios, alternating operations, squares, cubes, primes and other relationships. This page covers number series formulas, common concepts, solving methods and numerical examples used in quantitative aptitude.
What Is a Number Series?
For a series a₁, a₂, a₃, ..., each term may be obtained by adding, subtracting, multiplying or dividing by a value, or by applying a pattern involving powers and positions. For example, in 4, 9, 14, 19, the common difference is 5, so the next term is 24. A pattern may also alternate, as in 2, 5, 10, 13, 26, where the operations are +3, ×2, +3, ×2.
Number Series Video Lessons
Watch short topic-wise lessons for quick revision.
Special Sums: n, n² and n³ Formulas
Learn the formulas for the sums of the first n natural numbers, their squares, and their cubes, with a clear understanding of how these special summations are represented.
Number Series Formula & Tricks
Important Formulas
Use this when the difference between consecutive terms is constant. Here, a₁ is the first term, d is the common difference and n is the term position.
Use this when the ratio between consecutive terms is constant. Here, r is the common ratio.
Calculate consecutive differences. If the first differences are not constant, check second or higher differences.
Some series follow a direct rule based on the term position, such as aₙ = n² + 1 or aₙ = 2ⁿ.
Quick Tricks
Subtract each term from the next. If the differences are not constant, calculate the differences between those differences.
When consecutive terms do not show one rule, divide the series into odd-position and even-position terms.
If terms increase or decrease quickly, divide each term by its predecessor. A constant ratio indicates a geometric pattern.
Numbers close to squares, cubes, prime numbers or factorials often indicate a position-based pattern.
Number Series Concepts
Arithmetic Difference Series
If the common difference is d, each term is obtained by adding d to the previous term. For 18, 14, 10, 6, the difference is −4, so the next term is 2. The difference can be positive, negative or zero.
Multiplication and Division Series
For a constant-ratio series, divide a term by the preceding term. In 5, 15, 45, 135, the ratio is 3, so the next term is 405. If the multiplier changes, inspect the sequence of multipliers, such as ×2, ×3, ×4.
Difference of Differences
For 4, 9, 16, 25, 36, the first differences are 5, 7, 9 and 11. The second difference is 2, so the next first difference is 13 and the next term is 49. Higher-order differences can be checked if the second differences are not constant.
Alternate-Term Series
Write the terms in two rows or group positions 1, 3, 5 and 2, 4, 6. Each subsequence may use addition, subtraction, multiplication or another rule. This method is useful when the full sequence appears irregular.
Square, Cube and Position-Based Series
Common forms include n², n³, 2ⁿ, n² + c and n³ − c. Always check the starting position before selecting a rule. For 1, 4, 9, 16, the terms are 1², 2², 3² and 4².
Number Series Data Table
Use the structured data below while following the explanation and examples.
Common Number Series Patterns
Typical rules used to identify missing terms in number series questions.
| Pattern | Example | Rule | Next term |
|---|---|---|---|
| Constant difference | 6, 10, 14, 18 | +4 | 22 |
| Constant ratio | 3, 9, 27, 81 | ×3 | 243 |
| Increasing differences | 2, 5, 10, 17 | +3, +5, +7 | 26 |
| Alternating operations | 2, 5, 10, 13, 26 | +3, ×2 repeatedly | 29 |
| Squares plus one | 2, 5, 10, 17 | n² + 1 | 26 |
Note: For an alternating pattern, verify the operation sequence across all available terms before finding the next term.
Learn more concepts on PrepShots
Start ₹1 Trial →Practice Number Series Questions
Practise published questions related to this topic.
Want more practice questions?
Practice in App →Number Series Quick Quiz
Attempt 5 questions and check your score instantly.
Keep practising
Practice more Number Series questions in the PrepShots app and continue from your current topic.
Practice More Questions - Start ₹1 Trial →Quick Revision Notes
Number Series Revision Points
Use these checks while solving a number series.
- First calculate consecutive differences.
- If first differences vary, check second differences.
- For rapidly increasing terms, compare consecutive ratios.
- Separate odd and even positions when one common pattern is not visible.
- Check squares, cubes, primes, factorials and position-based expressions.
- A valid rule must explain every given term, not only the last two terms.
- In a decreasing series, retain the negative sign in differences or ratios.
- For a constant-difference series, use aₙ = a₁ + (n − 1)d.
Number Series FAQs
What is the formula for the nth term of an arithmetic number series?
The formula is aₙ = a₁ + (n − 1)d, where a₁ is the first term and d is the constant difference. For 5, 8, 11, the fourth term is 5 + 3 × 3 = 14.
How do you find the next term when the differences increase regularly?
Find the first differences and then examine their differences. In 4, 9, 16, 25, the differences are 5, 7 and 9; the next difference is 11, so the next term is 36.
How can alternate terms be checked in a number series?
List the odd-position terms separately from the even-position terms. In 2, 4, 6, 8, 18, 16, the odd and even subsequences must be checked independently; a single rule should not be assumed from adjacent terms.
What is the difference between an arithmetic and a geometric series?
An arithmetic series has a constant difference, while a geometric series has a constant ratio. For example, 4, 7, 10 has difference 3, whereas 4, 12, 36 has ratio 3.
How do you identify a square-based number series?
Compare each term with consecutive squares or a fixed adjustment from them. In 2, 5, 10, 17, the terms are 1² + 1, 2² + 1, 3² + 1 and 4² + 1.
What should be done when both differences and ratios look irregular?
Check alternate terms, changing operations, squares, cubes, primes and digit-based relationships. A proposed rule should reproduce all the given terms before it is used to find the missing term.