Basic math
SEQUENCES AND SERIES
Sequences: the topic of sequences is an aspect of mathematics that is widely treated in the field of real analysis, generally sequence of numbers is a set of numbers in a definite order of arrangement and formed according to a definite rule. the members of the sequence are called terms, therefore if the numbers of terms are finite ,it is called finite sequence. Otherwise infinite sequence.
For example, the page number of a textbook or novel is countable and that makes it a finite
sequence while the set ,… is an infinite sequence because it has no known ending. Series: A series is simply formed by sum of terms of the sequence. For example the sequence 1,3,5,7… is formed into a series as 1+3+5+7… The nth term of a sequence is a law of formation by which any term in the sequence may be obtained. This law is often given by means of a formula dependant on n such that the substitution of n=1, 2, 3, 4, 5… in the formula yields the first , second, third,….nth terms. The nth term is the general term of the sequence which is usually Written as an. the method of finding general terms in any sequence is simple. You just follow the three basic rules of “natural number terms, even terms, and odd terms.” (1)natural number terms: 1,2,3,4,5,… = an=n for all n=>1 (2)even terms: 2,4,6,8,… = an=2n for all n=>2 (3) odd terms: 1,3,5,7,9,…= an=(2n-1) for all n=>1 Examples 1: obtain the general terms of the following sequence. ,,,,… (a),,,,… (c),,,… Solution: (a) here the general term is quite simple in the sense that you will evaluate the numerator first before proceeding to the denominator. It can be seen that the numerator is a set of natural numbers beginning with 1,hence the general terms of the numerator is n, looking at the denominator we can see that it is a set of even numbers beginning with 2,so the general term of the denominator is 2n. hence the general term of the overall sequence is an = for all n=>1. The numerator of the sequence here is even set of number beginning with 2,so the general term is 2n,the denominator is a set of natural number beginning with 4 so the general term is n+3.hence the general term of the sequence is an = , for all n=>1 The numerator is an odd set of number beginning with 1,so the general term of the numerator is 2n-1,while the denominator is an even set of numbers beginning with 2,so the general term of the denominator is 2n,now the general of the sequence is an = , for all n=>1.. Finding the nth terms of the general sequence.. Example 2: find the 5th and 8th terms of the sequence whose nth terms are (a)2n+1 (b)3-5n. Solution: (a)5th term=2(5)+1=11 8th term=2(8)+1=17 (b) 5th term=3-5(5)=-22 8th term=3-5(8)=-37
sequence while the set ,… is an infinite sequence because it has no known ending. Series: A series is simply formed by sum of terms of the sequence. For example the sequence 1,3,5,7… is formed into a series as 1+3+5+7… The nth term of a sequence is a law of formation by which any term in the sequence may be obtained. This law is often given by means of a formula dependant on n such that the substitution of n=1, 2, 3, 4, 5… in the formula yields the first , second, third,….nth terms. The nth term is the general term of the sequence which is usually Written as an. the method of finding general terms in any sequence is simple. You just follow the three basic rules of “natural number terms, even terms, and odd terms.” (1)natural number terms: 1,2,3,4,5,… = an=n for all n=>1 (2)even terms: 2,4,6,8,… = an=2n for all n=>2 (3) odd terms: 1,3,5,7,9,…= an=(2n-1) for all n=>1 Examples 1: obtain the general terms of the following sequence. ,,,,… (a),,,,… (c),,,… Solution: (a) here the general term is quite simple in the sense that you will evaluate the numerator first before proceeding to the denominator. It can be seen that the numerator is a set of natural numbers beginning with 1,hence the general terms of the numerator is n, looking at the denominator we can see that it is a set of even numbers beginning with 2,so the general term of the denominator is 2n. hence the general term of the overall sequence is an = for all n=>1. The numerator of the sequence here is even set of number beginning with 2,so the general term is 2n,the denominator is a set of natural number beginning with 4 so the general term is n+3.hence the general term of the sequence is an = , for all n=>1 The numerator is an odd set of number beginning with 1,so the general term of the numerator is 2n-1,while the denominator is an even set of numbers beginning with 2,so the general term of the denominator is 2n,now the general of the sequence is an = , for all n=>1.. Finding the nth terms of the general sequence.. Example 2: find the 5th and 8th terms of the sequence whose nth terms are (a)2n+1 (b)3-5n. Solution: (a)5th term=2(5)+1=11 8th term=2(8)+1=17 (b) 5th term=3-5(5)=-22 8th term=3-5(8)=-37
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