Sequence and Series UPSC

Sequence and Series: One of the most significant concepts in Arithmetic is Sequence and Series. A sequence represents a collection of elements that can be repeated in any order. In simpler terms, a sequence is a systematic arrangement of objects or items in a sequential manner.

For instance: a1, a2, a3, a4, …

On the other hand, a series refers to the sum of all the elements within a sequence. In essence, a series is the cumulative total of all the elements present in a sequence. An example commonly used to explain sequence and series is Arithmetic Progression (AP).

For example: If a1, a2, a3, a4, … is considered a sequence, then the sum of terms in the sequence a1 + a2 + a3 + a4, … is considered a series.

Concept for Sequences:

Sequence: An ordered list of numbers following a certain pattern.
Term: Each individual number in a sequence.
Arithmetic Sequence: A sequence where the difference between consecutive terms is constant.
Geometric Sequence: A sequence where the ratio between consecutive terms is constant.
Fibonacci Sequence: A famous sequence where each term is the sum of the two preceding terms, starting with 0 and 1.
Recursive Formula: A formula that expresses a term in a sequence in terms of previous terms.
Explicit Formula: A formula that directly gives the value of the nth term in a sequence without relying on previous terms.

Concept for Series:

Series: The sum of the terms in a sequence.
Arithmetic Series: The sum of an arithmetic sequence.
Geometric Series: The sum of a geometric sequence.
Convergence: A series is said to converge if the sum of its terms approaches a finite value as more terms are added.
Divergence: A series is said to diverge if the sum of its terms does not approach a finite value.
Partial Sum: The sum of a finite number of terms in a series.
Infinite Series: A series that has an infinite number of terms.

Important Concepts:

Sum of an Arithmetic Series: S_n = (n/2) * (a + l), where S_n is the sum of the first n terms, a is the first term, and l is the last term.
Sum of a Geometric Series: S_n = a * (1 – r^n) / (1 – r), where S_n is the sum of the first n terms, a is the first term, r is the common ratio, and n is the number of terms.
Telescoping Series: A series where most terms cancel out, leaving only a few terms to compute the sum.
Alternating Series Test: A test used to determine whether an alternating series converges.
Ratio Test: A test for the convergence of a series based on the ratio of consecutive terms.
Integral Test: A test for the convergence of a series by comparing it to an integral.

These are just some of the keywords and concepts related to sequences and series in mathematics. They play a crucial role in various fields, including calculus, number theory, and mathematical analysis.

Types of Sequence and Series

Some of the most common examples of sequences are:

Arithmetic Sequences
Geometric Sequences
Harmonic Sequences
Fibonacci Numbers

Arithmetic Sequences

A sequence where each term is generated by adding or subtracting a fixed number to/from the previous term.

Geometric Sequences

A sequence where each term is obtained by multiplying or dividing the previous term by a constant value.

Harmonic Sequences

A series of numbers forms a harmonic sequence if the reciprocals of all its elements form an arithmetic sequence.

Fibonacci Numbers

Fibonacci numbers constitute an intriguing sequence in which each element is the sum of the two preceding elements, starting with 0 and 1. The sequence is defined as F0 = 0, F1 = 1, and Fn = Fn-1 + Fn-2.

Sequence and Series Formulas

The sequence of A.P: The n^th term a_n of the Arithmetic Progression (A.P) a, a+d, a+2d,…a, a+d, a+2d,… is given by

a_n=a+(n–1)d

Where,

a	First-term
d	Common difference
n	Position of the term
l	Last term

Arithmetic Mean: The arithmetic mean between a and b is given by A.M=a+b2

The sequence of G.P: The nth term a_n of the geometric progression a, ar, ar², ar³,…, is a_n=arⁿ–1an=ar^n–1

The geometric mean between a and b is G.M= ±\sqrt{ab}

Sequence of H.P: The n^th term a_n of the harmonic progression is a_n= 1a+(n–1)d

The harmonic mean between a and b is H.M=2aba+b

Series of A.P: If S_n denotes the sum up to n terms of A.P. a, a+d, a+2d,…a, a+d, a+2d,… then

S_n = n2(a+l),

S_n = n2[2a+(n–1)d]

The sum of n A.M between a and b is A.M = n(a+b)2

Series of G.P: If S_n denotes the sum up to n terms of G.P is S_n=a(1–rn)1–r; r≠1 and l=arⁿ

The sum S of infinite geometric series is S=a1–r;

Difference Between Sequences and Series

Let us find out how a sequence can be differentiated with series.

Sequences	Series
Set of elements that follow a pattern	Sum of elements of the sequence
Order of elements is important	Order of elements is not so important
Finite sequence: 1,2,3,4,5	Finite series: 1+2+3+4+5
Infinite sequence: 1,2,3,4,……	Infinite Series: 1+2+3+4+……

Example

Write an A.P when its first term is 10 and the common difference is 3.

Solution:

Step 1: Arithmetic Progression = A.P. = a, a+m , a+2m , a +3m, a+4m…….

Step 2: here, a=10 and m = 3

So let put its value in the equation

Step 3: 10, 10+3, 10 +2*3, 10 + 3*3, 10 + 4 * 3 ………

We get,

10,13,16, 19, 22, …….

Example 2

Write the first three terms of the sequence defined as a_n= n²+ 1

Solution:

Step1: we have a_n= n²+ 1

Step 2: Putting n = 1,2,3. We get

Step 3: a₁= 1²+ 1 = 1 + 1 = 2

a₂ = 2²+ 1 = 4 + 1 = 5

a₃= 3²+ 1 = 9 + 1 = 10

Quiz Time

Write the first five terms of each of the following sequence whose nth terms are:

a_n = 3n + 2
a_n = 3 n

Write an A.P. having 4 as the first term and -3 as the common difference.
Write a G.P. having 4 as the first term and 2 as the common ratio.

Summary Points

In a math series and sequence, the first term is denoted as “a,” the common difference is represented as “d,” and the nth term is identified as “an.”
The arithmetic sequence can be explained as a, a + d, a + 2d, a + 3d, …
Obtaining each term in a geometric progression involves multiplying the preceding term by the common ratio of the successive term.
The geometric progression formula is known as :

a_n = ar^n-1