Final answer:
To find the 13th term of an arithmetic sequence with the 7th term as 40 and the 10th term as 60, we first calculate the common difference, then the first term and finally use these values to determine the 13th term, which is 106.67.
Step-by-step explanation:
Finding the 13th Term of an Arithmetic Sequence
To find the 13th term of an arithmetic sequence when given the 7th and 10th terms, one needs to determine the common difference and then use it to calculate the requested term. The formula for the nth term of an arithmetic sequence is an = a1 + (n - 1)d, where a1 is the first term and d is the common difference.
Given: - The 7th term (a7) is 40. - The 10th term (a10) is 60.
We can set up two equations based on this formula:
1. 40 = a1 + 6d
2. 60 = a1 + 9d
By subtracting the first equation from the second, we find the common difference:
60 - 40 = (a1 + 9d) - (a1 + 6d)
20 = 3d d = 20 / 3
Using the value of d and the 7th term, we can find the first term (a1):
40 = a1 + 6(20 / 3) a1 = 40 - 40 / 3 a1 = 80 / 3
Finally, calculate the 13th term:
a13 = a1 + 12d a13 = (80 / 3) + 12(20 / 3) a13 = (80 + 240) / 3 a13 = 320 / 3 a13 = 106.67
Therefore, the 13th term of the arithmetic sequence is 106.67.