Sequence and series are used in mathematics as well as in our daily lives. A sequence is also known as progression and a series is developed by sequence. Sequence and series is one of the basic concepts in Arithmetic. Sequences are the grouped arrangement of numbers orderly and according to some specific rules, whereas a series is the sum of the elements in the sequence. For example, 2, 4, 6, 8 is a sequence with four elements and the corresponding series will be 2 + 4 + 6+ 8, where the sum of the series or value of the series will be 20.
There are various types of sequences and series depending upon the set of rules that are used to form the sequence and series. Sequence and series are explained in detail below.
The sequence is the group or sequential arrangement of numbers in a particular order or set of rules. Series is formed by adding the terms of a sequence. In a sequence, an individual term can be present in many places. Sequences can be of two types, i.e. infinite sequence and finite sequence and series will be then defined by adding the terms of the sequence. Sum of infinite terms in a series is possible in some cases as well.
Let us understand this with an example. 1, 3, 5, 7, 9, 11, ... is a sequence where there is a common difference of 2 between any two terms and the sequence goes on increasing up to infinity unless the upper limit is given. These types of sequences are known as arithmetic sequences. Now if we add the numbers in the sequence like 1 + 3 + 5 + 7+ 9... this will make a series of this sequence. These kinds of series are known as arithmetic series. A few examples of sequence and series are given in the image shown below:
Sequence and Series
Arithmetic Sequence → 10, 15, 20, 25, 30, 35,..............., un
Arithmetic Series → 10 + 15 + 20 + 25 + 30 + 35 + ............... + un
Geometric Sequence → 100, 50, 25,12.5, 6.25, 3.125,..., un
Geometric Series → 100 + 50 + 25 + 12.5 + 6.25 + 3.125 + . . . + un
The important differences between sequence and series are explained in the table given below:
There are various types of sequences and series, in this section, we will discuss some special and most commonly used sequences and series. The types of sequence and series are:
There are various formulas related to various sequences and series by using them we can find a set of unknown values like the first term, nth term, common parameters, etc. These formulas are different for each kind of sequence and series. Formulas related to various sequences and series are explained below:
Arithmetic Sequence and Series Formula
The various formulas used in arithmetic sequence are given below:
The various formulas used in geometric sequence are given below:
The following points are helpful to clearly understand the concepts of sequence and series.
Example 1: What will be the 15th term of the arithmetic sequence -3, -(1/2), 2…. using sequence and series formula?
Solution: Given a = -3, d = -(1/2) -(-3) = 5/2, n = 15
Using the formula for nth term of an arithmetic sequence: an = a+(n-1)d
Putting the known values:
a15 = -3 +(15-1) 5/2
a15 = 32
Example 2: Find the sum of the infinite geometric series -1 + 1/2 - 1/4 + 1/8 - 1/16 + ...
Solution: The common ratio of the given series is, r = -1/2.
Here, |r| = |-1/2| = 1/2 < 1.
Using the sequence and series formulas,
Sum of the given series = a / (1 - r)
= -1 / (1 - (-1/2))
= -1 / (3/2)
= -2/3
144 videos|100 docs|61 tests
|
|
Explore Courses for ACT exam
|