How to find the position of a term ?
Answers
Step-by-step explanation:
Work out the position to term rule for the following sequence: 5, 6, 7, 8, ... First, write out the sequence and the positions of each term. Next, work out how to go from the position to the term. In this example, to get from the position to the term, take the position number and add 4.
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Answer:
A sequence is an ordered set; usually the objects in the set (often numbers) are written with commas between them. e.g. 1,3,5,7 is a finite sequence, 1,3,5,7,... is an infinite sequence as is 2,4,8,16,... or 2,3,5,7,11,13,17,... (the primes.) The elements of the sequence are called terms; since the elements are ordered we can speak of the first term or a_1 , second term a_2 and the nth term a_n .
Sometimes there is a rule or rules that allow you to find terms of the sequence.
A term to term rule allows you to find the next number in the sequence if you know the previous term (or terms.) This is also called a recursive rule. For example, if the sequence is 1,3,5,7,... then in order to find the next term you add 2 to the previous term. a_5=a_4+2 or in general a_n=a_(n-1)+2 . The drawback to such a rule is that you have to know the previous terms. If I ask for the 100th term in this sequence, you must know the 99th term.
A position to term rule, also called an explicit rule, allows you to compute the value of any term. For the example 1,3,5,7,... the nth term is a_n=2n-1 . Thus the fifth term is 2(5)-1=9. The 100th term is 2(100)-1=199.
Sometimes finding a position to term rule is difficult. The Fibonacci sequence 1,1,2,3,5,8,13,... has as a term to term rule a_1=1,a_2=1, a_n=a_(n-2)+a_(n-1) for n>=3 .
The position to term rule is a_n=(((1+sqrt(5))/2)^n-((1-sqrt(5))/2)^n)/sqrt(5) .
Some sequences admit no known rule of either type, for instance the sequence of primes.