Arithmetic and Geometric Sequences and Series Formulas and Solved Examples
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 Complete Guide for Students
1. What is a sequence in mathematics?
A sequence is an ordered list of numbers that follow a specific rule or pattern. Each number in a sequence is called a term and is usually written as a₁, a₂, a₃, ... where the position matters. For example, in the sequence 2, 4, 6, 8, ... each term increases by 2. Sequences can be finite or infinite and are commonly defined using a formula for the nth term.
2. What is a series in mathematics?
A series is the sum of the terms of a sequence. If a sequence is 2, 4, 6, 8, then the corresponding series is 2 + 4 + 6 + 8. In notation, a series is written as S = a₁ + a₂ + a₃ + .... Series can be finite (limited terms) or infinite (infinitely many terms).
3. What is an arithmetic sequence?
An arithmetic sequence is a sequence in which the difference between consecutive terms is constant. This constant value is called the common difference (d).
- General formula: aₙ = a₁ + (n − 1)d
- Example: 3, 7, 11, 15, ... (d = 4)
4. What is the formula for the sum of an arithmetic series?
The sum of the first n terms of an arithmetic series is given by Sₙ = n/2 [2a₁ + (n − 1)d]. Alternatively, it can be written as:
- Sₙ = n/2 (a₁ + aₙ)
5. What is a geometric sequence?
A geometric sequence is a sequence where each term is obtained by multiplying the previous term by a constant called the common ratio (r).
- General formula: aₙ = a₁ rⁿ⁻¹
- Example: 2, 6, 18, 54, ... (r = 3)
6. What is the formula for the sum of a geometric series?
The sum of the first n terms of a geometric series is Sₙ = a₁(1 − rⁿ)/(1 − r) for r ≠ 1. For an infinite geometric series where |r| < 1, the sum is S = a₁/(1 − r). These formulas are essential in solving geometric series problems in algebra and calculus.
7. What is the difference between an arithmetic and a geometric sequence?
The key difference is that an arithmetic sequence uses a constant difference, while a geometric sequence uses a constant ratio.
- Arithmetic sequence: Add or subtract the same number each time.
- Geometric sequence: Multiply or divide by the same number each time.
8. How do you find the nth term of a sequence?
The nth term is found using a general formula based on the type of sequence.
- Arithmetic: aₙ = a₁ + (n − 1)d
- Geometric: aₙ = a₁ rⁿ⁻¹
9. What does it mean for a series to converge or diverge?
A series converges if its sum approaches a finite value, and diverges if it does not. For example:
- The infinite geometric series with |r| < 1 converges to a₁/(1 − r).
- The series 1 + 2 + 3 + 4 + ... diverges because its sum increases without limit.
10. What are sequences and series used for in real life?
Sequences and series are used to model patterns, growth, and financial calculations in real life. Common applications include:
- Compound interest (geometric series)
- Loan repayments and annuities
- Population growth models
- Physics and engineering calculations
