Common Types of Sequences and How to Identify Their Patterns
The arrangement of numbers in a particular order is known as a sequence. Sometimes, we are asked to find the value of a specific term in a sequence. One method to find the value of a specific term in a sequence is to extend the sequence until we reach the desired term, Next approach is to determine the sequence finding rule of the nth term of a sequence and then calculate the term you need.
Extending a sequence to find the missing term in a sequence is not always a realistic approach. For example, we cannot extend a sequence from the beginning to find a value 300th term. In this case, determining a sequence finding rule provides a more elegant and efficient way to find unknown terms in a sequence. Sequence rule finder is an online tool that enables you to find the unknown term in a sequence efficiently.
Term to Term Rule
A term to term rule enables you to find the next term in a given sequence if you know the previous term This is also known as a recursive rule. For example, if the given sequence is 2, 4, 6, 8, then to find the next term you can use the general formula: an = an-1 + 2 (a5 = a4 + 2). The drawback of the term-to-term rule is that you should know the previous term to calculate the next term.
General Rule of Arithmetic Sequence
Given a sequence with the first term a₁ and the common difference d, the nth or general rule of an arithmetic sequence is given by an = a1 + (n - 1)d.
Example:
Find the 10th term of an arithmetic sequence 5, 8,11, 13
a1 = 5 , d = (8 - 5) = 3
Accordingly.
a10 = 5 + (10 - 1)3
a10 = 5 + (9)3
a10 = 32
Explicit Rule
The explicit rule, also known as the position-to-term rule, allows you to calculate the value of any term. For example, in 2, 4, 6, 8…. the first term is 2, the second term is 4, the third term is 6, the fourth term is 8, and to calculate the fifth term here we use the formula an = 2n = 2(5) = 10. Hence, the fifth term here is 10. The 100th term is 2(100) = 200.
Solved Example
1. Write a rule for the nth term of the sequence given below.
1, 4, 9, 16, 25, 36,?
Solution:
Each term in the sequence follows the pattern 1² = 1, 2² = 4, 3² = 9, 4² = 16, 5² = 25…
These are the squares.
To find the nth term of a given sequence, we will follow the following rule.
Rule = an = n²
Here, an is “term number n”.
Accordingly, the 7th term of the given sequence is a7 = 7² = 49.
2. Find the next term of the sequence: 1, 2, 5, 10, 17,?
Solution:
In the given sequence, we can see each term is increasing. Taking the difference between adjacent terms we get:
2 - 1 = 1
5 - 2 = 3
10 - 5 = 5
17 - 10 = 7
Here, we can see the difference between the adjacent terms is odd numbers. Accordingly, the next term must be 9.
Hence, the next term in the sequence is 17 + 9 = 26.
FAQs on How to Find the Rule in Sequences: Step-by-Step Guide
1. What is a sequence in mathematics?
A sequence is an ordered collection of numbers, called terms, arranged according to a specific rule. For example, in the sequence 3, 6, 9, 12..., each term is obtained by adding 3 to the previous one. The position of each term is denoted by its index (1st, 2nd, 3rd, etc.).
2. How do you find a rule for a sequence?
To find a rule for a sequence, you should follow these steps:
Analyse the pattern: Look at the difference or ratio between consecutive terms.
Check for a common difference: If the difference is constant, it is an Arithmetic Progression (AP).
Check for a common ratio: If the ratio is constant, it is a Geometric Progression (GP).
Formulate the nth term: Use the standard formula for that type of sequence to express the rule, such as a + (n-1)d for an AP.
3. What are the main types of sequences students learn as per the CBSE syllabus?
As per the CBSE/NCERT curriculum for the 2025-26 session, the primary types of sequences studied are:
Arithmetic Sequence (or Progression, AP): A sequence where the difference between any two consecutive terms is constant.
Geometric Sequence (or Progression, GP): A sequence where the ratio of any two consecutive terms is constant.
Other important types often introduced include the Fibonacci Sequence and Harmonic Sequence.
4. As an example, how can you find the rule for the sequence 2, 5, 8, 11...?
First, find the difference between consecutive terms: 5 - 2 = 3, 8 - 5 = 3, and 11 - 8 = 3. Since there is a common difference of 3, this is an arithmetic sequence. The rule, or the formula for the nth term (a_n), is a_n = a + (n-1)d. Here, 'a' (the first term) is 2 and 'd' (the common difference) is 3. So, the rule is a_n = 2 + (n-1)3, which simplifies to a_n = 3n - 1.
5. What is the formula for the nth term of an Arithmetic Progression (AP)?
The formula to find the nth term (a_n) of an Arithmetic Progression (AP) is: a_n = a + (n-1)d. In this formula:
- a_n is the term you want to find.
- a is the first term of the sequence.
- n is the position of the term.
- d is the common difference between terms.
6. What is the formula for the nth term of a Geometric Progression (GP)?
The formula to find the nth term (a_n) of a Geometric Progression (GP) is: a_n = ar^(n-1). In this formula:
- a_n is the term you want to find.
- a is the first term of the sequence.
- r is the common ratio between terms.
- n is the position of the term.
7. Why is it important to find a general rule (nth term) for a sequence?
Finding a general rule, or the nth term, is important because it allows you to:
- Predict any term: You can calculate the value of any term (like the 100th term) without listing all the preceding ones.
- Understand the structure: The rule defines the fundamental nature and growth pattern of the sequence.
- Solve problems efficiently: It is the basis for finding the sum of a series and solving complex problems in algebra and calculus.
8. What is the key difference between an Arithmetic and a Geometric sequence?
The key difference lies in how their terms progress. In an Arithmetic Sequence, you add a constant value (the common difference, d) to get from one term to the next. In contrast, in a Geometric Sequence, you multiply by a constant value (the common ratio, r) to get from one term to the next. For example, 2, 4, 6, 8... is arithmetic (add 2), while 2, 4, 8, 16... is geometric (multiply by 2).
9. Can a sequence be both arithmetic and geometric?
Yes, a sequence can be both arithmetic and geometric, but only in one specific case: a constant sequence where all terms are the same (e.g., 5, 5, 5, 5...). For this sequence, the common difference is d = 0 (making it arithmetic), and the common ratio is r = 1 (making it geometric). Any non-constant sequence cannot be both.
10. What are some real-world applications of finding rules for sequences?
Sequences and their rules have many real-world applications. For instance:
- Finance: Calculating compound interest or loan repayments involves geometric sequences.
- Physics: The distance an object in free fall travels during successive seconds follows an arithmetic progression.
- Biology: The growth of a bacteria population under ideal conditions can be modelled using a geometric sequence.
- Computer Science: The complexity and performance of algorithms often rely on patterns that can be described as sequences.
