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1. To determine and simplify an expression that represents the sum of the first n terms of the series 5, 8, 11, 14, and so forth, one must first identify the nature of the series. This particular series is an arithmetic sequence where the first term, denoted as a, is 5, and the common difference, d, is 3. The general formula for the nth term of an arithmetic sequence can be expressed as a_n = a + (n – 1)d. By substituting the known values, the nth term can be formulated as a_n = 5 + (n – 1) * 3, which simplifies to a_n = 3n + 2.
2. Once the nth term has been established, the next step involves calculating the sum of the first n terms of the series. The formula for the sum of the first n terms, S_n, of an arithmetic series is given by S_n = n/2 * (a + a_n). In this case, substituting the values of a and a_n into the formula yields S_n = n/2 * (5 + (3n + 2)). This expression can be further simplified by combining the constants within the parentheses, resulting in S_n = n/2 * (3n + 7).
3. Finally, to express the sum in a more manageable form, one can multiply through by n/2. This leads to the final expression for the sum of the first n terms of the series, which is S_n = (3n^2 + 7n)/2. This formula provides a clear and concise representation of the sum, allowing for easy calculation of the total for any specified number of terms, n, in the series. Thus, the expression effectively encapsulates the cumulative value of the series up to the nth term.
