In Exercises 43–54, express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. 1+2+3+⋯+ 30

Verified step by step guidance

1. Identify the pattern in the series: 1, 2, 3, ..., 30.
2. Recognize that this is an arithmetic series where each term increases by 1.
3. The general term of the series can be expressed as a function of the index i: a_i = i.
4. Use summation notation to express the series: \( \sum_{i=1}^{30} i \).
5. This notation indicates that you are summing the terms from i = 1 to i = 30.

Key Concepts

Summation Notation

Summation notation is a mathematical shorthand used to represent the sum of a sequence of numbers. It typically uses the Greek letter sigma (Σ) to denote the sum, with an index of summation that indicates the starting and ending values. For example, Σ from i=1 to n represents the sum of all integers from 1 to n.

Index of Summation

The index of summation is a variable that represents the position of each term in the sequence being summed. In the expression Σ from i=1 to n, 'i' is the index that takes on integer values starting from 1 up to n. This index allows us to systematically add each term in the sequence, making it easier to express and compute sums.

Arithmetic Series

An arithmetic series is the sum of the terms of an arithmetic sequence, where each term increases by a constant difference. The sum of the first n natural numbers, such as 1 + 2 + 3 + ... + 30, can be calculated using the formula S_n = n/2 * (first term + last term). Understanding this concept helps in efficiently calculating sums without listing all terms.