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.
${\sum }_{i=1}^{17}(5i+3)$

Verified step by step guidance

1. Identify the arithmetic sequence given by the general term \(a_i = 5i + 3\), where \(i\) ranges from 1 to 17.
2. Find the first three terms by substituting \(i = 1, 2, 3\) into the formula: \(a_1 = 5(1) + 3\), \(a_2 = 5(2) + 3\), \(a_3 = 5(3) + 3\).
3. Find the last term by substituting \(i = 17\) into the formula: \(a_{17} = 5(17) + 3\).
4. Recall the formula for the sum of the first \(n\) terms of an arithmetic sequence: \(S_n = \frac{n}{2} (a_1 + a_n)\), where \(n\) is the number of terms, \(a_1\) is the first term, and \(a_n\) is the last term.
5. Substitute \(n = 17\), the first term \(a_1\), and the last term \(a_{17}\) into the sum formula to express the sum \(S_{17}\) without calculating the final value.

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. For example, in the sequence defined by 5i + 3, the difference between consecutive terms is constant (5). Understanding this helps identify the terms and their pattern.

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 process of adding many terms without listing them all.

Evaluating Summation Notation

Summation notation (Σ) represents the sum of terms defined by a formula over a range of indices. Here, it sums the expression 5i + 3 from i = 1 to 17. Understanding how to interpret and expand this notation is essential for finding specific terms and the total sum.