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Summation notation is a concise way of representing the sum of a sequence of numbers. It is expressed using the sigma symbol, \(\sum\). The expression beneath the sigma indicates the starting value of the index, while the expression above it indicates the ending value. For example, in the series \(\sum_{i=3}^{7}(5i+2)\), \(i\) starts at 3 and ends at 7. The expression \(5i+2\) is the formula used to generate each term in the series. By using summation notation, we can easily denote long sums without writing out all individual terms. This is particularly useful in mathematics and statistics. When calculating the series, we substitute each value of \(i\) into the formula and sum the results.

An arithmetic series is a sequence of numbers in which the difference between consecutive terms is constant. This difference is called the common difference. The series can be expressed in the form \(a, a+d, a+2d, a+3d, \ldots\) where \(a\) is the first term, and \(d\) is the common difference. For example, the series generated by the expression \(5i+2\) when \(i\) ranges from 3 to 7 forms an arithmetic series: \[ 17, 22, 27, 32, 37 \]. Here, \(a=17\), and the common difference \(d\) is 5. To find the sum of an arithmetic series, we can use the formula \[ S_n = \frac{n}{2} (a + l) \], where \(S_n\) is the sum of the series, \(n\) is the number of terms, \(a\) is the first term, and \(l\) is the last term.

A finite series is a sequence of numbers that has a definite number of terms. Unlike infinite series, which continue indefinitely, finite series have an endpoint. In the given exercise, the series \(\sum_{i=3}^{7}(5i+2)\) is finite because it sums only from \(i=3\) to \(i=7\). To solve a finite series, we often substitute each term one by one and then add them up. Finite series are easier to manage mathematically due to their limited number of terms. Understanding finite series is important because they appear frequently in algebra, calculus, and real-world applications. The total sum of a finite series can be found by either manually adding all the terms, as seen in the exercise, or by using appropriate formulas for specific types of finite series, such as arithmetic or geometric series.