algebra sequence

What are you talking about?

We have an arithmetic progression here, where a1 is the first term and the third term is one to be found:

the progression is descending as arithmetical ratio is -1/4, so the third term is 5584 - 2*1/4. We used here the formula for calculating the nth term of arithmetic progression that is an = a1 + (n - 1)r.

find a3 if a1=5584 r=-1/4

Well guest, is that what you meant.

It does sound reasonable, the notation that I am used to is a little different.
And, r usually stands for common ratio, which would make it a geometric progression.

I think the answer is more likely

a ₃= a ₁ * r ^(n-1)

a ₃ = 5584 * (-1/4) ²

a ₃= 349

Misunderstanding of this kind happens often solely because of absence of elementary skill to present a task.

As for notation I've got accustomed to one being used in the Russian math elementaries, theory and practices. For example any term of arithmetic progression is denoted as an with arithmetic ratio denoted as d. Geometric progression - bn and q respectively.