In Exercises 8–9, find each indicated sum. This is a summation, refer to the textbook.

Verified step by step guidance

1. Identify the type of summation given in the problem. It could be an arithmetic series, a geometric series, or another type of series.
2. Determine the general formula for the series. For an arithmetic series, use the formula for the sum of an arithmetic series: \( S_n = \frac{n}{2} (a_1 + a_n) \), where \( n \) is the number of terms, \( a_1 \) is the first term, and \( a_n \) is the last term. For a geometric series, use \( S_n = a_1 \frac{1-r^n}{1-r} \), where \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms.
3. Substitute the known values into the formula. Identify the first term, the last term (or the common ratio for geometric series), and the number of terms.
4. Simplify the expression by performing the arithmetic operations as indicated in the formula.
5. Verify your steps and ensure that all calculations are correct, and that the formula used is appropriate for the type of series given.

Key Concepts

Summation Notation

Summation notation, often represented by the Greek letter sigma (Σ), is a concise way to express the sum of a sequence of terms. It includes an index of summation, a lower limit, an upper limit, and a formula for the terms being summed. Understanding how to interpret and manipulate this notation is essential for calculating sums in algebra.

Arithmetic Series

An arithmetic series is the sum of the terms of an arithmetic sequence, where each term after the first is obtained by adding a constant difference. The formula for the sum of the first n terms of an arithmetic series is S_n = n/2 * (a + l), where a is the first term, l is the last term, and n is the number of terms. Recognizing this structure helps in efficiently calculating sums.

Geometric Series

A geometric series is the sum of the terms of a geometric sequence, where each term is found by multiplying the previous term by a constant ratio. The sum of the first n terms can be calculated using the formula S_n = a(1 - r^n) / (1 - r) for r ≠ 1, where a is the first term and r is the common ratio. Understanding geometric series is crucial for solving problems involving exponential growth or decay.