Answer:
[tex]\textsf{a)} \quad \textsf{Recursive Rule}: \quad a_n=5n-2[/tex]
[tex]\textsf{Explicit Rule}: \quad \begin{cases}a_1=3\\a_n=a_{n-1}+5\end{cases}[/tex]
b) 4998
c) explicit rule (see below for explanation)
d) see below
Step-by-step explanation:
An explicit formula for an arithmetic sequence allows you to find the nth term of the sequence.
A recursive formula for an arithmetic sequence allows you to find the nth term of the sequence provided you know the value of the previous term in the sequence.
Part a
Explicit Formula
[tex]a_n=a+(n-1)d[/tex]
where:
- [tex]a_n[/tex] is the nth term
- a is the first term
- n is the number of the term
- d is the common difference
Given sequence: 3, 8, 13, 18, 23, ...
To find the common difference, subtract one term from the next term.
Therefore:
Substituting the found values of a and d into the formula to create a explicit rule for the given sequence:
[tex]\implies a_n=3+(n-1)5[/tex]
[tex]\implies a_n=3+5n-5[/tex]
[tex]\implies a_n=5n-2[/tex]
Recursive Formula
[tex]a_n=a_{n-1}+d \quad \textsf{for }n\geq 2[/tex]
where:
- [tex]a_n[/tex] is the nth term
- [tex]a_{n-1}[/tex] is the term immediately before the nth term
- d is the common difference
We already know the common difference our previous calculations.
Therefore:
[tex]\implies a_n=a_{n-1}+5[/tex]
When giving a recursive rule we also have to define the first term of the sequence, as it is not part of the formula. Therefore, the full recursive rule for the given sequence is:
[tex]\begin{cases}a_1=3\\a_n=a_{n-1}+5\end{cases}[/tex]
Part b
To find the 1000th term of the sequence, use the explicit rule and substitute n = 1000:
[tex]\implies a_{1000}=5(1000)-2[/tex]
[tex]\implies a_{1000}=5000-2[/tex]
[tex]\implies a_{1000}=4998[/tex]
Therefore, the 1000th term of the sequence is 4998.
Part c
We used the explicit rule to find the 1000th term as this rule allows us to find the nth term of the sequence without having to know any previous terms.
Part d
The disadvantage of a recursive rule when compared to an explicit rule is that we need to know the previous term to find the nth term, whereas the explicit rule doesn't need this information.
Learn more about sequences here:
https://brainly.com/question/27924553
https://brainly.com/question/27775450
