In Exercises 29–42, find each indicated sum. 6Σi=1 5i

Verified step by step guidance

1. Identify the summation notation: \( \sum_{i=1}^{6} 5i \). This means you need to find the sum of the expression \( 5i \) as \( i \) goes from 1 to 6.
2. Write out the terms of the sum by substituting each integer value of \( i \) from 1 to 6 into the expression \( 5i \).
3. Calculate each term: \( 5 \times 1, 5 \times 2, 5 \times 3, 5 \times 4, 5 \times 5, 5 \times 6 \).
4. List the results of each multiplication: \( 5, 10, 15, 20, 25, 30 \).
5. Add all the terms together to find the total sum: \( 5 + 10 + 15 + 20 + 25 + 30 \).

Key Concepts

Summation Notation

Summation notation, represented by the Greek letter Sigma (Σ), is a concise way to express the sum of a sequence of terms. The notation Σ indicates that you will sum a series of values, where the index (i in this case) runs from a specified lower limit to an upper limit. For example, 6Σi=1 5i means to sum the values of 5i for i starting from 1 up to 6.

Arithmetic Series

An arithmetic series is the sum of the terms of an arithmetic sequence, where each term increases by a constant difference. In the given problem, the terms being summed are generated by multiplying a constant (5) by the index (i), resulting in an arithmetic sequence. Understanding how to calculate the sum of such sequences is essential for solving summation problems.

Evaluating a Series

To evaluate a series, you need to substitute the index values into the expression and compute the sum of the resulting terms. In this case, you would calculate 5(1), 5(2), 5(3), 5(4), 5(5), and 5(6), and then add these products together. Mastery of this process is crucial for accurately finding the total sum indicated by the summation notation.