Respuesta :
The rule that can be used to find the next term of the sequence 27, 40, 53, 66, 79,... is given by: Option C: a_n = a_(n-1) + 13
What is arithmetic sequence?
An arithmetic sequence is sequence of integers with its adjacent terms differing with one common difference.
If the initial term of a sequence is 'a' and the common difference is of 'd', then we have the arithmetic sequence as:
[tex]a, a + d, a + 2d, ... , a + (n+1)d, ...[/tex]
Its nth term is
[tex]T_n = a + (n-1)d[/tex]
(for all positive integer values of n)
And thus, the common difference is
[tex]T_{n+1} - T_n[/tex]
for all positive integer values of n
Here, the considered sequence is 27, 40, 53, 66, 79,...
We see that:
40-27=13
53-40=13
and so on, the difference between each consequtive pair of terms of this sequence is 13.
Thus, due to having common difference in consequtive terms, this is an arithmetic sequence.
For this series, we've got:
- Initial term = a = 27
- Common difference = d = 13
Thus, its nth term is given by:
[tex]T_n = a+(n-1)d= 27 +(n-1)13 = 13n + 14[/tex]
Since the options are using the notation [tex]a_n[/tex] instead of [tex]T_n[/tex], so we will use that too. Thus, nth term and (n-1)th terms are:
[tex]a_n = 13n + 14\\a_{n-1} = 13(n-1) + 14 = 13n + 1\\[/tex]
Thus, we can write nth term in the form of (n-1)th term as:
[tex]a_n = 13n + 14 = 13n + 1 + 13 = a_{n-1} + 13\\\\a_n = a_{n-1} + 13[/tex]
Thus, the rule that can be used to find the next term of the sequence 27, 40, 53, 66, 79,... is given by: Option C: a_n = a_(n-1) + 13
Learn more about arithmetic sequence here:
https://brainly.com/question/3702506
