Write out the first three terms and the last term. Then use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum.
${\sum }_{i=1}^{30}(-3i+5)$

Verified step by step guidance

1. Identify the arithmetic sequence given by the general term: \(a_i = -3i + 5\) for \(i = 1, 2, \ldots, 30\).
2. Find the first three terms by substituting \(i = 1, 2, 3\) into the formula: \( a_1 = -3(1) + 5 \), \( a_2 = -3(2) + 5 \), \( a_3 = -3(3) + 5 \).
3. Find the last term by substituting \(i = 30\) into the formula: \( a_{30} = -3(30) + 5 \).
4. Use the formula for the sum of the first \(n\) terms of an arithmetic sequence: \( S_n = \frac{n}{2} (a_1 + a_n) \) where \(n = 30\), \(a_1\) is the first term, and \(a_n\) is the last term.
5. Substitute the values of \(n\), \(a_1\), and \(a_{30}\) into the sum formula to express the sum \(S_{30}\), then simplify the expression to find the sum.

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a list of numbers where each term after the first is found by adding a constant difference to the previous term. In this problem, the sequence is defined by the formula a_i = -3i + 5, which generates terms by substituting i = 1, 2, 3, etc.

Sum of the First n Terms of an Arithmetic Sequence

The sum of the first n terms of an arithmetic sequence can be found using the formula S_n = n/2 * (a_1 + a_n), where a_1 is the first term and a_n is the nth term. This formula simplifies the addition of many terms by using only the first and last terms.

Evaluating Terms from a Given Formula

To find specific terms in the sequence, substitute the term number i into the formula a_i = -3i + 5. For example, the first three terms are found by plugging in i = 1, 2, and 3. The last term corresponds to i = 30, as given in the summation.