What is the motivation behind the study of sequences? I was discussing some ideas with my professor and he always says that before you work on something in mathematics, you need to know the motivation for studying/working on it.
A better way to put this would be,

Why were sequences studied and sought after?

 A: I don't think it is that clear cut. Motivation is important (you don't want to work on a topic nobody cares about) but sometimes the motivations is just that (a sizable number of) people finds it interesting!
Of course you have more chances of finding interested people if the topic "pops up" in various problem, so people think it's worth to study it 
Anyhow, I think sequences are very interesting in their own right. For example a very natural question is "What is the sum of the first $n$ natural numbers? What about the first $n$ squares? " after finding the answer for that one, one starts to think "hey what if $n \to \infty$? " what is going to happen?
Sequences are in a way a very natural generalization of finite sums ;-) and results like $$ \sum \frac 1{n^2}=\frac{\pi^2}6$$
certainly add to the "charm " of the topic 
If you really want to find a more 'practical' motivation, sequences pop up all the time in physics, probability, etc. The most famous example is probably the Zeno's paradox, where the distance between Achilles and the turtle is always halving. Zeno concluded that Achilles will never reach the turtle, but had he studied sequences he would have recognized that $$\sum \frac 1{2^n} = 2\neq \infty$$
A: Sometimes with a sequence of simple objects you can approximate an object that is complicated.
The sequence $\{x_n:n\ge1\}$ defined by $x_1=1$ and
$$
x_{n+1}=\frac12\Bigl(x_n+\frac2{x_n}\Bigr)
$$
converges to $\sqrt 2$ as $n\to\infty$ (see here). So you have a sequence of rational numbers that converges to an irrational number.
A: Relations between causes and effects is a central topic in all branches of science. In quantitative mathematics, it is embodied by the concept of a function. For instance, you might be interested by the relation between the length of an iron rod and the temperature, $L=f(T)$.
An interesting special case of functions are those having the set of natural numbers, $\mathbb N$, as their domain. Such functions are the sequences. Depending on the situations, the argument $n$ can be seen as time events, as the index of an iteration, as a number of steps from an initial state, or anything that can be enumerated in a "uniform" way. They correspond to the universe of discrete phenomena as opposed to continuous ones. (Like the amount of $CO_2$ in the atmosphere as a function of the date.)
The study of sequences allows to draw general properties, like "can we compare two sequences ?", and focuses on the questions around infinity: "what can we say about the trend of a given sequence, when we let the argument grow forever ?"
A: I think that one of the most interesting thing about sequences is that sequences are "easy" to manage.
In a first place, when you modeling something, a recurrence relation often arise. When you have this relation, you can compute (by hand or with a computer) all the terms you want.
But in a lot of cases, your phenomenon is continuous (e.g. physical movement), hence the most appropriate way to model that is (real) functions. However, it is hard to deal with functions $\mathbb R \to \mathbb R$. So you discretize your problem  using sequences (e.g. Euler method to solve ODE).
It is the same in a more abstract point of view: when you are doing some topology, the spaces can be very very difficult to manage. For example, the set of the continuous functions over $[0,1]$: here, manipulate sequences of functions is easier than work with big (non countable) bundle  of functions. 
So we try to turn properties concerning "open sets", "compactness", etc. into properties concerning sequences as much as we can: for example in a metric space, a set $A$ is closed iff every sequence converging converges in $A$ (which is often simpler to use than "$A$ is closed iff its complementary is open").
That's why it is important to be fluent with sequences! If you are not, it will be difficult to understand well more abstracts concepts (since those concepts use sequences a lot).
A: Zeno's Paradox.
Zeno noted that before you can travel to a place, you must first travel halfway to the place, then from there travel half the rest of the way, etc.  He reasoned that movement is impossible because you cannot travel an infinite number of steps in a finite time.
In the language of sequences, the relevant numbers to Zeno's paradox are: $1/2$, $3/4$, $7/8$, etc.  His final statement is, in effect, that this infinite series that is strictly increasing cannot converge to a finite limit.
Study of sequences has shown that this last statement is patently false.  It is perfectly possible to have an infinite sequence of increasing numbers that converges to a finite value.  We even determined methods of proving which sequences do converge and what their final values are.
From there, we have many other practical uses for sequences, like determine the values of $\pi$ or $e$ to as much precision is necessary for specific applications.
A: I think the most important take-home here is something that the answers so far haven't emphasized enough - by chaining together a bunch of very simple 'stuff', you can build amazingly complex stuff, but you're still left with components that are very easy to work with. 


*

*Suppose you want to differentiate $f(x) e^{\arctan{\sqrt{1+\ln x}}}$ or some other annoying function. This is tough. But differentiating $f(x) = \sum a_n x^n$ is easy.

*Suppose you want to do some form of numerical computation at all relating to geometry. You'd probably need $\pi$. $\pi$ is an incredibly complicated beast, consisting of an infinitely long sequence of digits that has no predictability whatsoever. It's incomprehensibly complex. But if express it as a chain of simple terms (in the form of a sequence!), we can approximate it to arbitrary precision. 
We are essentially exchanging one very annoying object for a lot of simple objects. But when is this allowed? Here's where convergence is the key - which is exactly what we study in those seemingly pointless root ratio test exercises they throw at you.
A: Once you have a collection of objects (be it numbers or other abstract entities) it's natural to overlay structure onto those objects (e.g. a topology, an ordering, an algebra of sets, etc.) We can also arrange them into a sequence. So we want to study sequences in general and to understand their properties in a variety of given contexts. Usually, we start with sequences of numbers. This is a good place to start since data is often a list of numbers.
Any set of data is essentially a sequence of finite length. I consider it natural to ask the question: "If extrapolated into the future for all 'time', what would the data 'look like'?" Ignoring the math needed to extrapolate data into the future, the question that arises is how to understand the behavior of the (now infinite) list of data. So we need to know all the different types of sequences and how they behave. 
A: Sequences can be thought of as a reasonably general class of patterns, or the kind of data that would be used to measure properties of a more complicated pattern that is not itself a sequence.   Considering how much of mathematics and science is a study of patterns, one would expect sequences to appear almost everywhere that mathematics is used.
