Sum of an Arithmetic Sequence

Find the sum of $3n + 2$ from $n = 1$ to $n = 5$.

This problem involves finding the sum of a sequence described by the linear function $3n + 2$ over a specified range. This is a common exercise in the study of arithmetic sequences, where each term increases by a constant amount. In this specific instance, the sequence begins at $n = 1$ and ends at $n = 5$. When approaching problems like these, it's important to recognize the linear structure of the sequence.

This is an arithmetic sequence, because the term difference (also known as the common difference) is constant. Identifying this allows you to apply the formula for the sum of an arithmetic sequence, which is useful in simplifying calculations. The formula takes into account the number of terms in the sequence, their common difference, and the first term.

In general, understanding how to derive and work with arithmetic sequences is a foundational skill in discrete mathematics, often used when dealing with sequences and series more broadly. Mastery of these concepts can also be applied to more complex scenarios involving series, such as geometric series or when integrating calculus concepts into discrete structures. Recognizing arithmetic patterns and implementing the formula for their sums makes such problems more manageable and provides a robust tool for solving a wide variety of mathematical and practical problems.