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An arithmetic sequence is a list of numbers in which each term is obtained by adding a fixed number, called the common difference, to the previous term. It's like climbing or descending a staircase where each step is the same height—the common difference in this analogy is the height of each step.

Let's visualize with a simple example. Think of the sequence as 2, 4, 6, 8, and so on. Each term after the first is 2 more than the term before it. This '2 more' is the common difference. Each arithmetic sequence can be described by two key elements: the first term and the common difference. As we climb this numerical staircase, no matter how high we go (or how far along the sequence we are), we're always stepping with the same rhythm—that is the essence of an arithmetic sequence.

The common difference is what gives an arithmetic sequence its name and its uniformity. It's the steady beat to which the sequence marches. This difference can be positive, negative, or even zero, which leads to interesting sequences. In our previous sequence, 2, 4, 6, 8... the common difference was positive (+2). However, let's take another sequence such as 10, 5, 0, -5... Here, each term decreases by 5, hence the common difference is negative (-5).

Identifying the common difference is straightforward—pick any term in the sequence (after the first one), subtract the term before it, and voilà, you have the common difference. This crucial piece allows us to predict the future of the sequence and also to formulate a general term for it.

Now, to unleash the true power of an arithmetic sequence, we aim to find any term, say the 100th or even the 1000th, without listing all the preceding terms. This is where the formula for the nth term of an arithmetic sequence comes into play: \(a_n = a_1 + (n-1) * d\).

In the context of the provided sequence 10, 5, 0, -5, -10..., once we've identified the first term (\(a_1 = 10\)) and the common difference (\(d = -5\)), we plug these values into the formula to calculate any term in the sequence (\(a_n\)). For instance, if you want to find the 20th term, plug in \(n = 20\): \(a_{20} = 10 + (20-1) * (-5)\), simplifying to \(a_{20} = 10 - 95 = -85\).

This formula is a reliable shortcut and an essential tool for finding terms deep within an arithmetic sequence without breaking a sweat.