Given An Arithmetic Sequence In The Table Below, Create The Explicit Formula And List Any Restrictions To The Domain.$\[ \begin{tabular}{|l|l|} \hline $n$ & $a_n$ \\ \hline 1 & 40 \\ \hline 2 & 47 \\ \hline 3 & 54 \\ \hline \end{tabular} \\]A.
**Arithmetic Sequence Explicit Formula and Domain Restrictions** ===========================================================
What is an Arithmetic Sequence?
An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. This constant difference is called the common difference.
Given Arithmetic Sequence
| n | a_n |
|---|---|
| 1 | 40 |
| 2 | 47 |
| 3 | 54 |
Step 1: Find the Common Difference
To find the common difference, we need to subtract any term from its previous term. Let's subtract the second term from the first term:
a_2 - a_1 = 47 - 40 = 7
So, the common difference is 7.
Step 2: Find the First Term
The first term is already given in the table: a_1 = 40.
Step 3: Write the Explicit Formula
The explicit formula for an arithmetic sequence is given by:
a_n = a_1 + (n - 1)d
where a_n is the nth term, a_1 is the first term, n is the term number, and d is the common difference.
Substituting the values we found earlier, we get:
a_n = 40 + (n - 1)7
Simplifying the formula, we get:
a_n = 40 + 7n - 7
a_n = 33 + 7n
Explicit Formula
a_n = 33 + 7n
Domain Restrictions
Since the term number n is a positive integer, the domain of the sequence is all positive integers.
Q&A
Q: What is the value of the 5th term?
A: To find the value of the 5th term, we need to substitute n = 5 into the explicit formula:
a_5 = 33 + 7(5) = 33 + 35 = 68
Q: What is the common difference?
A: The common difference is 7.
Q: What is the first term?
A: The first term is 40.
Q: What is the explicit formula?
A: The explicit formula is a_n = 33 + 7n.
Q: What are the domain restrictions?
A: The domain restrictions are all positive integers.
Conclusion
In this article, we found the explicit formula and domain restrictions for a given arithmetic sequence. We also answered some common questions related to arithmetic sequences.
Frequently Asked Questions
Q: What is an arithmetic sequence?
A: An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant.
Q: How do I find the common difference?
A: To find the common difference, subtract any term from its previous term.
Q: How do I write the explicit formula?
A: The explicit formula for an arithmetic sequence is given by: a_n = a_1 + (n - 1)d
Q: What are the domain restrictions for an arithmetic sequence?
A: The domain restrictions for an arithmetic sequence are all positive integers.
Q: How do I find the value of a term in an arithmetic sequence?
A: To find the value of a term in an arithmetic sequence, substitute the term number into the explicit formula.
Q: What is the explicit formula for an arithmetic sequence?
A: The explicit formula for an arithmetic sequence is given by: a_n = a_1 + (n - 1)d
Q: What is the common difference for an arithmetic sequence?
A: The common difference for an arithmetic sequence is the constant difference between any two consecutive terms.
Q: What is the first term for an arithmetic sequence?
A: The first term for an arithmetic sequence is the first number in the sequence.
Q: What are the domain restrictions for an arithmetic sequence?
A: The domain restrictions for an arithmetic sequence are all positive integers.