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One of the fundamental concepts in arithmetic sequences is the 'common difference', often denoted as 'd'. This term is the constant amount each element in the sequence increases or decreases by, from one term to the next. In the given exercise, the common difference is 5. This means that each term is 5 more than the previous term. Understanding the common difference is crucial because it allows us to predict the rest of the sequence once we know at least one of its terms.

For example, if we start with a term, let's say 10, and know that the common difference is 5, the sequence would look like this: 10, 15, 20, 25, and so forth. Each term is simply the previous term plus the common difference. If, conversely, the common difference were -3, our sequence would decrease by 3 each time: 10, 7, 4, 1, and so on. Recognizing this pattern is key to solving arithmetic sequence problems and finding rules that define them.

The arithmetic sequence formula is a mathematical expression that defines the specific value of any term in an arithmetic sequence. The formula is given as \[a_{n} = a_{1} + (n-1)d\], where:\

\
• \(a_{n}\) is the nth term of the sequence,\
• \(a_{1}\) is the first term of the sequence,\
• \(n\) is the term number,\
• \(d\) is the common difference.\

\Using this formula, we can calculate the value of any term in the sequence once we know the first term and the common difference. To apply it successfully, we often need to work backward if we're given a term other than the first. In our exercise, we knew the 9th term (\(a_{9}=55\)) and used it along with the common difference to find the first term (\(a_{1}\)). This approach illustrates the flexibility of the arithmetic sequence formula and how it can be manipulated to solve for unknown values within the sequence.

The terms in a sequence are the individual elements that make up the sequence. In arithmetic sequences, each term is derived from the previous one by adding the common difference. Knowing any term in a sequence, especially the first term (\(a_{1}\)), and the common difference allows us to find all the other terms. In the exercise, after finding the first term, the rule for determining any term in the sequence becomes achievable. In general, the nth term is calculated by starting with the first term and adding the common difference multiple times, dependent on the position of the term we are looking for. That is, for the nth term, the common difference is added n-1 times. Hence, the sequence in our exercise can be written as \[a_{n} = 15 + (n-1)\times5\], which represents a rule where by substituting different values of n, we get corresponding terms of the sequence. It implies a systematic approach: given a particular term number, we can plug it into our established rule to find the value of that term without needing to list out the entire sequence.