Sequences and Series Meaning
Focused on dealing with a list of objects and events (in a certain fashion), sequences and series teach students all about different functions that dominate a sequence chain. Further, sequences and sets also provide a description of different operations that are used for adding different quantities of a sum. The chapter Sequences and Series Class 11 is about mathematical concepts that are based around lines and series. Students can refer to the CBSE Class 11 Sequence and Series study resource on Vedantu to the to get a complete idea of the basic calculations of Sequence and Series at its core.
About Sequence and Series
In mathematics, a sequence is a compile list of objects or events that are ordered or placed in a sequential fashion, created in such a way that each member of a given sequence comes before or after every other member in the list of sequences.
More precisely, a sequence is a mathematical expression where the function paired to a domain is equal to the set of positive integers.
Key Terms in Sequences and Series
Sequence: An aggregative function whose domain is a subset of natural numbers. A sequence can often be denoted in a fashion: 1, 2, 3,… ,n, as f1, f2, f3, …., fn , where fn = f(n).
Real Sequence: Sequence where the range is a subset of R is called a real sequence.
Series: In the mathematical expression where a1, a2, a3 , … , it is a sequence and the expression a1 + a2 + a3 + … + an is a series.
Progression: Sequence where there are certain rules to follow, becomes a progression.
Finite Series: A series that is cued with a finite number of terms is called finite series.
Infinite Series: A series that comprises an infinite number of terms is called infinite series.
Solved Examples of Sequence and Series
Listed below is a compilation of different types of sums that you may come across in the chapter, Sequences and Series. It is a composition of different types and levels of sums from the chapter.
Q1. Between any two numbers ‘a’ and ‘b’, show how the ‘n’ numbers can be inserted, in such a formation that the resulting sequence is an Arithmetic Progression.
Solution:
Let, the sequence start with the following progression:
B1, B2, B3,……, Bn. Let the suggested sequence have ‘n’ numbers, in such a manner that between b and c, C1, C2, C3,……, Cn, c is in A.P.
Here, b is the 1st term and b is (n+2)th term. Therefore,
c = b + d[(n + 2) – 1] = b + d (n + 1).
Hence, common difference (d) = (c - b)/(n + 1)
Now, A1 = b + d = b + ((c - b)/(n + 1))
B2 = b + 2d = b + ((2(c - b)/(n + 1))
Bn = b + nd= b + ((n(c - b)/(n + 1))}
Therefore, the nth term around the suggested geometric progression is highlighted with the help of
= arn-1
Sum of the nth term:
Sn = n/2 [2b + (n-1)d]
In the solution stated above, n denoted the number of terms, b denoted the first term and d is about the common difference.
Q2. Derive the first five terms of the given terms where:
an = n (n + 2)
Solution:
Given,
nth term of a sequence an = n (n + 2)
By substituting n = 1, 2, 3, 4, and 5, we get the first five terms
a1 = 1(1 + 2) = 3
a2 = 2(2 + 2) = 8
a3 = 3(3 + 2) = 15
a4 = 4(4 + 2) = 24
a5 = 5(5 + 2) = 35
Therefore, the required terms are 3, 8, 15, 24, and 35.
FAQs on Sequences and Series
1. What is the fundamental difference between a sequence and a series?
A sequence is an ordered list of numbers, called terms, that follow a specific rule or pattern. For example, 2, 4, 6, 8,... is a sequence where each term is 2 more than the previous one. A series, on the other hand, is the sum of the terms of a sequence. Using the same example, the corresponding series would be 2 + 4 + 6 + 8 + ... . In essence, a sequence is the list, while a series is the sum of that list.
2. What are the main types of progressions a student studies in Class 11 Maths?
In the CBSE Class 11 syllabus for 2025-26, students primarily focus on three main types of progressions:
- Arithmetic Progression (A.P.): A sequence where the difference between any two consecutive terms is constant. This constant value is called the common difference (d).
- Geometric Progression (G.P.): A sequence where the ratio of any two consecutive terms is constant. This constant value is known as the common ratio (r).
- Harmonic Progression (H.P.): A sequence is in H.P. if the reciprocals of its terms are in A.P. While less detailed, its relationship with A.P. is a key concept.
3. How can you determine if a given sequence is an Arithmetic Progression (A.P.) or a Geometric Progression (G.P.)?
To identify the type of progression, you must check the relationship between consecutive terms. For an Arithmetic Progression, subtract each term from its succeeding term (e.g., term 2 - term 1, term 3 - term 2). If this difference is constant throughout the sequence, it is an A.P. For a Geometric Progression, divide each term by its preceding term (e.g., term 2 / term 1, term 3 / term 2). If this ratio is constant, it is a G.P.
4. Can a series with an infinite number of terms have a finite sum? Explain how.
Yes, an infinite series can have a finite sum, but only under specific conditions. This is a key concept for an infinite Geometric Progression (G.P.). The sum to infinity of a G.P. converges to a finite value if and only if the absolute value of its common ratio 'r' is less than 1 (i.e., |r| < 1). The formula for this sum is S = a / (1 - r), where 'a' is the first term. If |r| ≥ 1, the terms do not get smaller, and the sum will be infinite.
5. What is the importance of the relationship between Arithmetic Mean (A.M.) and Geometric Mean (G.M.)?
The relationship between the Arithmetic Mean (A.M.) and Geometric Mean (G.M.) of two positive numbers is a fundamental inequality in mathematics: A.M. ≥ G.M. This concept is important because it is used to find the maximum or minimum values of functions and expressions without using calculus. For instance, if the sum of two numbers is constant, their product is maximum when the numbers are equal. This has practical applications in optimisation problems in fields like engineering and economics.
6. How are sequences and series applied in real-world scenarios like finance?
Sequences and series have many practical applications, especially in finance. For example, compound interest calculations use the formula for the amount in a Geometric Progression, where the principal amount grows by a constant ratio each period. Similarly, calculating the total value of regular investments over time (an annuity) involves finding the sum of a G.P. This helps in planning for retirement, loans, and other long-term financial goals.
7. What are 'special series', and why are their sum formulas important?
As per the NCERT syllabus, 'special series' refer to series whose terms do not form a standard A.P. or G.P. but follow a specific algebraic pattern. The formulas for the sum of the first 'n' terms of these series are crucial tools for advanced mathematics. Key examples include:
- The sum of the first n natural numbers: Σn = n(n+1)/2
- The sum of the squares of the first n natural numbers: Σn² = n(n+1)(2n+1)/6
- The sum of the cubes of the first n natural numbers: Σn³ = [n(n+1)/2]²
These formulas are essential for calculus, physics, and engineering to sum up quantities that change in a predictable, non-linear way.
8. What makes a sequence 'recursive', and how does it differ from an explicit formula?
A sequence is defined recursively when each term is generated from one or more preceding terms. To define it, you need the initial term(s) and a rule to find the next term. For example, the Fibonacci sequence (1, 1, 2, 3, 5, ...) is recursive. In contrast, an explicit formula allows you to calculate any term directly using its position 'n' in the sequence, without needing to know the previous terms. For an A.P., a_n = a + (n-1)d is an explicit formula.
