Today our topic of discussion is Finite Series.
Finite Series
Finite Series
The term ‘order’ is widely used in our day to day life. For example, the concept of order is used to arrange the commodities in the shops, to arrange the events of drama and ceremony, to keep the commodities in attractive way in the godown.
Again, to make many tasks easier and attractive, we use large to small, child to old, light to heavy etc. types of order. Mathematical series have been originated from these concepts of order. In this chapter, the relation between sequence and series and contents related to them have been presented.
At the end of this chapter, the students will be able to-
‣ describe the sequence and series and determine the differences between them.
‣ explain finite series.
‣ form formulae for determining the fixed term of the series and the sum of fixed numbers of terms and solve mathematical problems by applying the formulae.
‣ determine the sum of squares and cubes of natural numbers.
‣ solve mathematical problems by applying different formulae of series.
‣ construct formulae to find the fixed term of a geometric progression and sum of fixed numbers of terms and solve mathematical problems by applying the formulae.
Sequence
Let us note the following relation:
Here, every natural number n is related to twice the number 2n. That means, the set of positive even numbers (2, 4, 6, is obtained by a method from the set of natural numbers (1, 2, 3,}. This arranged set of even number is a sequence
Hence, some quantities are arranged in a particular way such that the antecedent and subsequent terms becomes related. The set of quantities arranged in such a way is called a sequence.
The aforesaid relation is called a function and written as f(n) = 2n The general term of this sequence is 2n. The number of terms of any sequence is infinite. The way of writing the sequence with the help of general term is:
{2n}, n = 1, 2, 3 ,…or,{2n}+ ∞ n=1 or,{ 2n}
The first quantity of the sequence is called the first term, the second quantity is called second term, the third quantity is called the third term etc. The first term of the sequence 1, 3.5, 7 ,*** is 1 , the second term is 2, etc. Followings are the four examples of sequence:
1, 2, 3 ,***,n,….
1,3,5,***, 2n – 1 ….
1, 4, 9 ,***,n²…
1/2, 2/3, 3/4 ,***, n/n+1 ….
Series
If the terms of a sequence are connected successively by sign, a series is obtained. Such as, 1 + 3 + 5 + 7 +*** is a series. The differences between any two successive terms of the series are equal. Again, 2 + 4 + 8 + 16 +*** is a series. The ratio of two successive terms is equal. Hence, the characterstic of any series depends upon the relation between its two successive terms. Among the series, two important series are arithmetic series and geometric series.
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