Problem 47 - Algebra and Trigonometry

Question

express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. $$ 1+2+3+\dots+30 $$

Step-by-Step Solution

Answer

The given series in summation notation is \( \Sigma_ {i=1} ^ {30} i \)

Step 1: Understand the series

Given the arithmetic series, it is noticeable that the series starts from 1 and ends at 30 with a common difference of 1.

Step 2: Express in Summation Notation

The series can be expressed in summation notation with the lower limit being 1, the upper limit being 30, and the summation term \(i\), which represents the \(i^{th}\) term of the series. Therefore, the required expression in summation notation is : \( \Sigma_ {i=1} ^ {30} i \)

Key Concepts

Arithmetic Series

An arithmetic series is a collection of numbers that are added together, with each number differing from the previous one by a consistent amount. This difference is known as the common difference. For instance, in the arithmetic series 1, 2, 3, down to 30, the common difference is 1. This means each number increases by 1 from the previous number. Arithmetic series are useful because they allow us to quantify the total sum by applying a formula, particularly when the number of terms is large. Here, the formula used is: \[ S_n = \frac{n}{2} (a + l) \] 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. Knowing this helps us validate our sum, ensuring no term was missed.

Series Representation

Series representation refers to expressing a sequence of numbers in a concise mathematical form. In many cases, especially with arithmetic series, these can be very long or infinite. Summation notation is a powerful tool for this purpose, allowing us to write complex sums succinctly. The Greek capital letter \( \Sigma \), pronounced "sigma," indicates the sum of a sequence. Following the \( \Sigma \) symbol:

• The expression \( i=1 \) is the lower limit, indicating where the summation starts.
• The upper limit, 30 in this case, tells us the last number to include in the sum.
• The term after the \( \Sigma \) symbol, \( i \), is the general term of the series, representing each step or element to be summed.

By using summation notation, arithmetic series are represented in a compact form: \( \Sigma_{i=1}^{30} i \). This representation is not only mathematical but also simplifies tasks like identifying the sum.

Index of Summation

The index of summation is a crucial part of understanding series and their notation. It serves as a counter or variable that represents each term's position in the sequence as you sum the series. In our example \( \Sigma_{i=1}^{30} i \), the index 'i' plays this role. A few things to remember about the index of summation:

• It starts at the lower limit, here 1, and increments up to the upper limit, in this case, 30.
• The index is often a variable, commonly 'i', but it can be any letter.
• It shows how each element is calculated and processed in the sequence.
• The complete expression \( i \) inside summation notation describes the sequence being summed.

Understanding the index of summation helps in comprehending how the total value of the series is computed. It's the backbone that organizes the series, making summation notation a powerful way to represent and calculate sums efficiently without listing every term individually.