Why use a+(n-1)d when we already have y=mx+b? I just wanted to know if there was some big difference between them? Otherwise, I don't really see a point in using completely different variable names and different-looking equations(which ultimately mean the same thing)? It just seems confusing and unnecessary to me.
Thanks!
 A: Technically both will lead you to the same equation, take the sequence
a+(n-1)d
3, 7, 11, 15, 19
The common difference = +4 while the initial term, a is 3
3 + (n-1)4 = 3 + 4n - 4 
= 4n - 1

From this equation, you can gauge that the first term is -1+4 = 3
y=mx+b
m = (7-3) / (2-1) = 4
3 = (4 * 1) + b
b = -1

y = 4x - 1

The difference is that arithmetic sequences don't usually have decimal/fractional n, they are usually integers. You always find the 1st term or the 2nd term or the 5th term, you'd never find the 2.5th term. Whereas for y=mx+b equations, you can plug in any x.
Moreover, you know for a fact that arithmetic sequences have a starting term (a), but y=mx+b go on forever. Of course the y-intercept would be the starting point but as said before, y=mx+b can compute for any real x.
I think that both equations can work to solve the same thing but the first one is more rigid in its approach and shows that it is a arithmetic sequence where n has to be an integer.
Hopefully this helps.
A: I think that notation can be a big source of confusion for students, and it is important for teachers, and especially book authors, to carefully choose the best notation.
On the other hand, when you learn about functions in high school, you may get so used to seeing $y = f(x)$ that it is very confusing the first time you see something like $z = g(t)$. Depending on how far you go in your studies, you should eventually become comfortable with the fact that any letter or other symbol could be chosen to name a quantity, function, or other object. Often we use many different letters and symbols so that we do not use the same symbol for two different objects. Sometimes our choice of letter is based on what that quantity actually represents, like $E$ for energy or $t$ for time.
When students first learn sequences, often the $n^\mathrm{th}$ term in the sequence is represented by $a_n$. We often choose the symbol $n$ for a quantity that is meant to be an integer. Would you prefer instead that we call the $x^\mathrm{th}$ term in the sequence $y_x$?
Often a sequence is defined by a recursion equation and intial condition, such as $a_1 = 20$ and $a_{n+1} = a_n + d$.  I suspect that $d$ is chosen to stand for the common "difference" between terms, but that's just a guess.
The analytic solution to an arithmetic recursion equation is given by $a_n = a_1 + (n-1)d$
If you would prefer that this be written as $y = mx+b$, then how would you prefer the recursion equation to be written? $b = 20$, $y_0 = b$, and $y_{x+1} = y_x + m$?
Indeed, that would give the analytic solution $y_x = mx + b$
I suspect though that making the equation look more similar to $y = mx + b$ would actually lead to more confusion, since the $y$ still has the subscript $x$, and since $x$ can only take on integer values. I suspect that making the notation for sequences different from $y = mx + b$ probably helps students keep them distinct in their minds, especially when they are first learning about them.
That said, it is good to make the connection between the two.
