Sequence And Series
Sequence and series is one of the basic topics in Arithmetic. An itemized collection of elements in which repetitions of any sort are allowed is known as a sequence, whereas series is the sum of all elements. An arithmetic progression is one of the common examples of sequence and series.
- In short, a sequence is a list of items/objects which have been arranged in a sequential way.
- A series can be highly generalized as the sum of all the terms in a sequence. However, there has to be a definite relationship between all the terms of the sequence.
The fundamentals could be better understood by solving problems based on the formulas. They are very similar to sets but the primary difference is that in a sequence, individual terms can occur repeatedly in various positions. The length of a sequence is equal to the number of terms and it can be either finite or infinite. This concept is explained in a detailed manner in Class 11 Maths. With the help of definition, formulas and examples we are going to discuss here the concepts of sequence as well as series.
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Table of contents:
Sequence and Series Definition
A sequence is an arrangement of any objects or a set of numbers in a particular order followed by some rule. If a_{1}, a_{2}, a_{3}, a_{4},……… etc. denote the terms of a sequence, then 1,2,3,4,…..denotes the position of the term.
A sequence can be defined based on the number of terms i.e. either finite sequence or infinite sequence.
If a_{1}, a_{2}, a_{3}, a_{4,} ……. is a sequence, then the corresponding series is given by
S_{N} = a_{1}+a_{2}+a_{3} + .. + a_{N}
Note: The series is finite or infinite depending if the sequence is finite or infinite.
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 in which every term is created by adding or subtracting a definite number to the preceding number is an arithmetic sequence.
Geometric Sequences
A sequence in which every term is obtained by multiplying or dividing a definite number with the preceding number is known as a geometric sequence.
Harmonic Sequences
A series of numbers is said to be in harmonic sequence if the reciprocals of all the elements of the sequence form an arithmetic sequence.
Fibonacci Numbers
Fibonacci numbers form an interesting sequence of numbers in which each element is obtained by adding two preceding elements and the sequence starts with 0 and 1. Sequence is defined as, F_{0} = 0 and F_{1} = 1 and F_{n} = F_{n-1} + F_{n-2}
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Sequence and Series Formulas
List of some basic formula of arithmetic progression and geometric progression are
Arithmetic Progression | Geometric Progression | |
Sequence | a, a+d, a+2d,……,a+(n-1)d,…. | a, ar, ar^{2},….,ar^{(n-1)},… |
Common Difference or Ratio | Successive term – Preceding term
Common difference = d = a_{2} – a_{1} |
Successive term/Preceding term
Common ratio = r = ar^{(n-1)}/ar^{(n-2)} |
General Term (nth Term) | a_{n} = a + (n-1)d | a_{n} = ar^{(n-1)} |
nth term from the last term | a_{n} = l – (n-1)d | a_{n} = 1/r^{(n-1)} |
Sum of first n terms | s_{n} = n/2(2a + (n-1)d) | s_{n} = a(1 – r^{n})/(1 – r) if r < 1
s_{n} = a(r^{n} -1)/(r – 1) if r > 1 |
*Here, a = first term, d = common difference, r = common ratio, n = position of term, l = last term
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+…… |
Sequence and Series Examples
Question 1: If 4,7,10,13,16,19,22……is a sequence, Find:
- Common difference
- nth term
- 21st term
Solution: Given sequence is, 4,7,10,13,16,19,22……
a) The common difference = 7 – 4 = 3
b) The nth term of the arithmetic sequence is denoted by the term T_{n} and is given by T_{n} = a + (n-1)d, where “a” is the first term and d, is the
common difference.
T_{n} = 4 + (n – 1)3 = 4 + 3n – 3 = 3n + 1
c) 21st term as: T_{21} = 4 + (21-1)3 = 4+60 = 64.
Question 2: Consider the sequence 1, 4, 16, 64, 256, 1024….. Find the common ratio and 9th term.
Solution: The common ratio (r) = 4/1 = 4
The preceding term is multiplied by 4 to obtain the next term.
The nth term of the geometric sequence is denoted by the term T_{n} and is given by T_{n} = ar^{(n-1)}
where a is the first term and r is the common ratio.
Here a = 1, r = 4 and n = 9
So, 9th term is can be calculated as T_{9} = 1* (4)^{(9-1)}= 4^{8} = 65536.
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Frequently Asked Questions
What does a Sequence and a Series Mean?
A sequence is defined as an arrangement of numbers in a particular order. On the other hand, a series is defined as the sum of the elements of a sequence.
What are Some of the Common Types of Sequences?
A few popular sequences in maths are:
- Arithmetic Sequences
- Geometric Sequences
- Harmonic Sequences
- Fibonacci Numbers
What are Finite and Infinite Sequences and Series?
Sequences: A finite sequence is a sequence that contains the last term such as a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}……a_{n. }On the other hand, an infinite sequence is never-ending i.e. a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}……a_{n….}.
Series: In a finite series, a finite number of terms are written like a_{1 }+ a_{2 }+ a_{3} + a_{4 }+ a_{5} + a_{6 + }……a_{n}. In case of an infinite series, the number of elements are not finite i.e. a_{1 }+ a_{2 }+ a_{3} + a_{4 }+ a_{5} + a_{6 + }……a_{n }+_{…..}