Introduction :
The number sequence is defined as a function generally that formed from the set of natural numbers to the set of real numbers.If the sequence is denoted by the letter z, then the image of n ∈ N which may be the natural number, under the sequence z is z(n) = z_{n}. Since the domain for every sequence is the set of natural numbers, the images of 1, 2, 3, … n … under the number sequences are denoted by z_{1}, z_{2}, z_{3} … z_{n}, … respectively. Here z_{1}, z_{2}, z_{3} … z_{n}, … form the sequence.
The sequence is generally represented by its range.
Recursive formula:
The recursive formula is generally defined as the sequence which may be described by specifying its few terms first and it is the general formula for finding the sequence of other terms in the way of its preceding terms.
For example, 11, 14, 15, 19, 24, …, is a sequence because each term (except the first two) is obtained by taking the sum of preceding two terms.
For that sequence,the corresponding recursive formula is z_{n} + 2 = z_{n} + z_{n+1} , n ≥ 1 here z_{1}=11, z_{2}= 14
Terms of a sequence:
The various numbers occurring in a sequence are generally called as its terms. Here,we denote the terms of a sequence by z_{1}, z_{2}, z_{3}, … , z_{n}, … , the subscripts denote the position of the term. The n^{th} term is called the general term of the sequence.
For example, consider the given sequence 21, 23, 25, 27, … 2n − 1, … the 1st term is 21, 2nd term is 23, … … and nth term is 2n − 1.
Solution:
Given, let z_{n} = (− 1)^{n + 1} `((n+1)/n)`
substituting n = 7, we get
z_{7} = (− 1)^{7 + 1} `((7+1)/7)`
z_{7} = (− 1)^{8} `((8)/7)`
z_{7} = `((8)/7)`
For solving the number sequences we have some general forms,
For example if we have to solve(1 + 5)^{−1}
then,
we have the formula of (1 + x)^{−1} = 1 − x + x^{2} − x^{3} + …
(1 + 5)^{−1} = 0.167
But its expansion of the number sequences are used in some problems that require more derivation or it have the elimination than it is expressed in the form of the sequences.
Between, if you have problem on these topics Estimate Quotients, please browse expert math
related websites for more help on Rules Writing Numbers.
Problem : Find the 7th term of the sequence whose nth term is (− 1)^{n + 1} (n+1/n).
Solution:
Given a_{n} = (− 1)^{n + 1}(n+1/n)
substituting n = 7, we get
a_{7} = (− 1)^{7 + 1}(8/7)
a_{7} = 8/7.