Question

How do you find the formula for `a_n` for the arithmetic sequence `a_1=-4, a_5=16`?

Answer 1

Set up and solve a system of equations using the definition of the arithmetic sequence. Answer: `a_n=-4+5(n-1)`

Explanation:

We know that from the general formula for arithmetic sequences:
`a_n=a_1+(n-1)d`
where `a_n` is the nth term, `a_1` is the 1st term, `n` is the term number, and `d` is the difference between each term.

So, we write out what we know:
`a_1=-4`
`a_5=16`
Using the general formula for arithmetic sequences, the second equation becomes:
`a_5=a_1+(5-1)d=16`
By substituting `a_1=-4` into the above equation, we get:
`-4+4d=16`
Now, we simply solve for d:
`-1+d=4`
`d=5`

Therefore, the formula for the given arithmetic sequence is:
`a_n=-4+5(n-1)`