Answer:
The common difference d is d = -3
The recursive formula is: ![\mathbf{a_n=a_{n-1}-3}](https://tex.z-dn.net/?f=%5Cmathbf%7Ba_n%3Da_%7Bn-1%7D-3%7D)
Step-by-step explanation:
We need to find the common difference and the recursive formula.
a. 22, 19, 16, 13, …
First we will find common difference
The formula of arithmetic sequence is: ![a_n=a_1+(n-1)d](https://tex.z-dn.net/?f=a_n%3Da_1%2B%28n-1%29d)
where a_n is nth term, a_1 is 1st term and d is the difference
Looking at the sequence we get: a_1=22, a_2=19
Using these values we can find d, the common difference
![a_n=a_1+(n-1)d\\put\:n=2, a_2=19, a_1=22\\19=22+(2-1)d\\19=22+d\\d=19-22\\d=-3](https://tex.z-dn.net/?f=a_n%3Da_1%2B%28n-1%29d%5C%5Cput%5C%3An%3D2%2C%20a_2%3D19%2C%20a_1%3D22%5C%5C19%3D22%2B%282-1%29d%5C%5C19%3D22%2Bd%5C%5Cd%3D19-22%5C%5Cd%3D-3)
So, the common difference d is d = -3
Now, we will find the recursive formula:
The recursive formula is of type: ![a_n=a_{n-1}+d](https://tex.z-dn.net/?f=a_n%3Da_%7Bn-1%7D%2Bd)
We have found common difference d = -3
So, the recursive formula will be:
![a_n=a_{n-1}+d\\a_n=a_{n-1}+(-3)\\a_n=a_{n-1}-3](https://tex.z-dn.net/?f=a_n%3Da_%7Bn-1%7D%2Bd%5C%5Ca_n%3Da_%7Bn-1%7D%2B%28-3%29%5C%5Ca_n%3Da_%7Bn-1%7D-3)
The recursive formula is: ![\mathbf{a_n=a_{n-1}-3}](https://tex.z-dn.net/?f=%5Cmathbf%7Ba_n%3Da_%7Bn-1%7D-3%7D)