Answer:
A. -36
Explanation:
Step 1: Create a system of equations:
The formula for finding the nth term of an arithmetic sequence is given by:
an = a1 + (n - 1)d, where
- a1 is the first term,
- n is the term position (e.g., 40th or 1st),
- and d is the common difference.
Before we can find a40, we'll first need to know a1 and d.
Since we're only given a8 = 60 and a12 = 48, we can use a system of equations to first find both a1 and d.
We can start by plugging in the numbers we have to get two equations:
Plugging in 60 for an and 8 for n:
60 = a1 + (8 - 1)d
60 = a1 + 7d
Plugging in 48 for an and 12 for n:
48 = a1 + (12 - 1)d
48 = a1 + 11d
Method to solve: Substitution:
We can solve by substitution by isolating one of the variables.
First, let's isolate a1 in the first equation:
(60 = a1 + 7d) - 7d
-7d + 60 = a1
Now we can substitute -7d + 60 = a1 for a1 in 48 = a1 + 11d to solve for d, the common difference:
48 = -7d + 60 + 11d
(48 = 4d + 60) - 60
(-12 = 4d) / 4
-3 = d
Thus, the common difference is -d.
Now we can plug in -3 for d in 60 = a1 + 7d to solve for a1, the first term:
60 = a1 + 7(-3)
(60 = a1 - 21) + 21
81 = a1
Thus, the first term is 81.
Check the validity of the answer:
WE can check that our answers for d and a1 are correct by plugging in -3 for d and 81 for a1 in both equations and seeing if we get 60 and 48 on both sides of the equation:
Plugging in -3 for d and 81 for a1 in 60 = a1 + 7d:
60 = 81 + 7(-3)
60 = 81 - 21
60 = 60
Plugging in -3 for d and 81 for a1 in 48 = a1 + 11d:
48 = 81 + 11(-3)
48 = 81 - 33
48 = 48
Thus, our answers for d and a1 are correct.
Finally, we can find a40 by plugging in 81 for a1, 40 for n, and -3 for d:
a40 = 81 + (40 - 1)(-3)
a40 = 81 + 39 * -3
a40 = 81 - 117
a40 = -36
Thus, the answer is A. -36.