The first term of an arithmetic sequence is -5, and the tenth term is 13. Find the common difference.
The common difference d has been added 9 times to get to the 10th term, so
d = (13-(-5))/9 = 2
To find the common difference of an arithmetic sequence, you can use the formula:
\[a_n = a_1 + (n-1)d\]
where:
\(a_n\) is the nth term of the sequence,
\(a_1\) is the first term,
\(n\) is the term number,
\(d\) is the common difference.
Given that the first term (\(a_1\)) is -5, and the tenth term is 13 (\(a_{10}\)), we can substitute these values into the formula:
\[13 = -5 + (10-1)d\]
Now simplify the equation:
\[13 = -5 + 9d\]
Add 5 to both sides:
\[18 = 9d\]
Finally, divide both sides by 9 to solve for \(d\):
\[d = \frac{18}{9} = 2\]
Therefore, the common difference of the arithmetic sequence is 2.
To find the common difference in an arithmetic sequence, we can use the formula:
nth term = first term + (n - 1) * common difference,
where nth term is the term number we want to find, first term is the starting term, n is the position of the term, and common difference is the difference between two consecutive terms.
In this problem, we are given the first term, which is -5, and the tenth term, which is 13.
Using the formula, we can plug in the values:
13 = -5 + (10 - 1) * common difference.
To simplify the equation, we perform the necessary calculations:
13 = -5 + 9 * common difference.
Now, we need to isolate the common difference on one side of the equation. Let's subtract -5 from both sides of the equation:
13 + 5 = 9 * common difference.
18 = 9 * common difference.
To further isolate the common difference, we divide both sides of the equation by 9:
common difference = 18 / 9.
Finally, simplifying further:
common difference = 2.
Therefore, the common difference in this arithmetic sequence is 2.