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.
$\sum_{i = 1}^{30} (- 3 i + 5)$

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Identify the arithmetic sequence given by the general term: $a_i = -3i + 5$ for $i = 1, 2, \ldots, 30$.

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 $.

Find the last term by substituting $i = 30$ into the formula: $ a_{30} = -3(30) + 5 $.

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.

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.

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주요 개념

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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.

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.∑i=130(−3i+5)\(\sum\)_{i=1}^{30} (-3i + 5)