Determine the number of terms in the following arithmetic sequence 315,304,293,..... ,18.
The common difference between terms in an arithmetic sequence is obtained by subtracting any term from its subsequent term. So, we obtain $315-304=304-293=\cdots=33-18=15$. Listing every term in this sequence would be tedious, so we note that $315=18+15(19)$: the common difference has been applied 19 times, so there are $\boxed{20}$ terms.
To determine the number of terms in an arithmetic sequence, we need to find the common difference and use the formula:
Number of terms = (last term - first term) / common difference + 1
In this case, the first term is 315, and the last term is 18. We also need to find the common difference.
The common difference can be found by subtracting any two consecutive terms in the sequence. Let's subtract the second term (304) from the first term (315).
Common difference = 315 - 304 = 11
Now we can plug the values into the formula:
Number of terms = (18 - 315) / 11 + 1
Simplifying this equation:
Number of terms = -297 / 11 + 1
Number of terms = -27 + 1
Number of terms = -26
Since the number of terms cannot be negative, it means there must be an error in the question or sequence provided. Please double-check the sequence and make sure it is complete, or provide additional information if there is any missing.
To determine the number of terms in an arithmetic sequence, you need to know the first term, the common difference, and the last term. With this information, you can use the formula for finding the nth term of an arithmetic sequence and solve for n.
The first term of the sequence is 315, the common difference is the difference between any two consecutive terms, which in this case is -11 (304 - 315 = -11), and the last term is 18.
The formula for finding the nth term of an arithmetic sequence is:
an = a1 + (n - 1) * d
where:
an is the nth term,
a1 is the first term,
n is the number of terms,
d is the common difference.
In this case, we need to find the value of n. So we rewrite the formula as:
18 = 315 + (n - 1) * (-11)
Next, we simplify the equation:
18 = 315 - 11n + 11
18 = 326 - 11n
We can further simplify by subtracting 326 from both sides:
-308 = -11n
Finally, to solve for n, we divide both sides by -11:
n = 28
Therefore, there are 28 terms in the given arithmetic sequence.