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An arithmetic sequence is a list of numbers with a distinctive feature: each term after the first is derived by adding a consistent number, known as the common difference, to the previous term. The formula to find the nth term of an arithmetic sequence is given by:
\[ a_n = a_1 + (n-1) \times d \]
In this formula, \(a_n\) represents the nth term you're looking for, \(a_1\) is the first term in the sequence, \(n\) represents the term number, and \(d\) is the common difference between terms. For instance, if we want to determine the 5th term and we know the first term is 2 and the common difference is 3, we use the arithmetic sequence formula like this: \(a_5 = 2 + (5-1) \times 3\), giving us an answer of 14.

The common difference is a key element in arithmetic sequences. It's the engine behind the consistent increase or decrease from one term to the next that characterizes these sequences. Calculating the common difference, denoted as \(d\), is simple when you're given two terms in the sequence that are not next to each other:
\[d = \frac{a_m - a_n}{m-n}\]
In this formula, \(a_m\) corresponds to the later term and \(a_n\) to the earlier one, while \(m\) and \(n\) denote their respective positions in the sequence. Taking terms 12 and 27 from our exercise as an example, the common difference is computed as \(d = \frac{15 - 9}{27 - 12}\), resulting in a common difference of \(\frac{6}{15}\), which simplifies to \(\frac{2}{5}\). This value is uniform throughout the sequence.

To uncover the starting point of an arithmetic sequence, the first term \(a_1\), it's crucial when only later terms in the sequence are known. After determining the common difference, you can backtrack to find the first term. Here's how the operation looks mathematically:
\[a_1 = a_n - (n-1) \times d\]
Where \(a_n\) is a known term from the sequence, specifically not the first term, hence why \(n\) is greater than 1, and \(d\) is our already established common difference. Based on our initial problem if we choose the 12th term (9) and the common difference (\(\frac{2}{5}\)), the calculation for the first term would be: \(a_1 = 9 - (12-1) \times \frac{2}{5} = 9 - 22 \times \frac{2}{5} = 9 - 8.8 = 0.2\). Now, with the first term discovered, we can construct the whole sequence or find any term within it.