n
The fifth term of an arithmetic sequence is 23 and the 12th term is 72
1.2 What is the value of the tenth term?
To find the value of the tenth term, we can use the formula for the nth term of an arithmetic sequence:
a_n = a_1 + (n - 1)d,
where a_n is the nth term, a_1 is the first term, n is the position of the term, and d is the common difference.
We are given that the fifth term (a_5) is 23 and the 12th term (a_12) is 72. Let's use this information to find the common difference (d) and the first term (a_1).
Using the formula for the fifth term, we have:
a_5 = a_1 + (5 - 1)d
23 = a_1 + 4d
Using the formula for the 12th term, we have:
a_12 = a_1 + (12 - 1)d
72 = a_1 + 11d
Now we have a system of two equations with two variables (a_1 and d). We can solve this system to find the values.
First, let's subtract the first equation from the second equation to eliminate a_1:
(72 - 23) = (a_1 + 11d) - (a_1 + 4d)
49 = 7d
d = 49/7
d = 7
Now that we know the value of the common difference (d), we can substitute it back into one of the equations to find the first term (a_1). Let's use the first equation:
23 = a_1 + 4d
23 = a_1 + 4(7)
23 = a_1 + 28
a_1 = 23 - 28
a_1 = -5
Now that we know the first term (a_1 = -5) and the common difference (d = 7), we can find the value of the tenth term (a_10) using the formula:
a_10 = a_1 + (10 - 1)d
a_10 = -5 + (10 - 1)(7)
a_10 = -5 + 9(7)
a_10 = -5 + 63
a_10 = 58
Therefore, the value of the tenth term is 58.