Sum of First 40 Terms of an Arithmetic Sequence
Calculate the sum of the first 40 terms of the arithmetic sequence with a first term of 3 and a common difference of 2.
Calculating the sum of the first 40 terms of an arithmetic sequence involves understanding a few key mathematical concepts. An arithmetic sequence is defined by its first term and a common difference, which is the amount that each term increases or decreases relative to the previous term. One of the fundamental properties of an arithmetic sequence is that the difference between consecutive terms remains constant. In this problem, recognizing that pattern and employing it to determine the sum up to the 40th term is crucial. To solve this problem, you'll utilize the formula for the sum of an arithmetic sequence, which leverages the number of terms, the first term, and the last term or the common difference.
Strategically, it's important to break down the given information: the first term, the common difference, and the number of terms to find a generalized formula that helps in swift calculations. Furthermore, this problem reinforces a typical scenario where you apply these formulaic approaches to real-world problems or other mathematical challenges dealing with sequences. Understanding these concepts builds a foundation for more complex topics that involve series and sequences in calculus, preparing you for more advanced study in topics like infinite series and convergence.