Introduction to Sequences and Series
An itemized collection of elements in which repetitions of any sort is allowed is known as a sequence. They are very similar to sets and the members of the sequence are called elements (or terms). The length of a sequence is equal to the number of terms and it can be either finite or infinite. The primary difference between a sequence and a set is that in a sequence, individual terms can occur repeatedly in various positions.
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.
Table of Contents:
Sequence and Series Definition
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, where 1,2,3,4,…..denotes the position of the term.
A sequence can be defined based upon 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 = \(\sum_{n=1}^N\) a_n = a_1+a_2+a_3 + .. + a_N
Note: The series is finite or infinite depending upon the sequence is finite or infinite.
Sequence and Series Types
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 subtraction 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)
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 Ratio | Successive term – Preceding term
Example, a_2 – a_1 |
Successive term/Preceding term
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(r^{n-1})/(r-1) if r ≠ 1
s_n = na 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
Below are some common differences between sequences and 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 Questions
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 Tn and is given by Tn = a + (n-1)d, where “a” is the first term and d, is the common difference.
c) 21st term as: T21 = 4 + (21-1)3 = 4+60 = 64.
Question 2: Consider the sequence 1,4,16,64,256,1024….. Find: 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 Tn 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 T9 = 1* (4)^{(9-1)}= 4^{8} = 65536. Answer!
Articles Related to Series and Sequences:
Arithmetic Sequence Formula | Sum of Arithmetic Sequence Formula |
Difference Between Sequence and Series | Sequence Calculator |
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