Problem 43 - College Algebra

Question

Express each sum using summation notation. Use I as the lower limit of summation and i for the index of summation. $$1^{2}+2^{2}+3^{2}+\dots+15^{2}$$

Step-by-Step Solution

Answer

The sum \(1^{2}+2^{2}+3^{2}+\dots+15^{2}\) can be expressed in summation notation as \(\sum_{i=1}^{15}i^{2}\).

Step 1: Identify the pattern and establish the general term

Given the series: \(1^{2}+2^{2}+3^{2}+\dots+15^{2}\). Each term in the series represents a square of a positive integer starting from 1 and the series end at 15. This gives us the generic term of the series as \(i^{2}\) where i is a positive integer.

Step 2: Identify the limits of the series

The series starts with \(1^{2}\) which means that the lower limit, I, is 1. The last term is \(15^{2}\) which implies that the upper limit of the series is 15.

Step 3: Write the series in summation notation

The final step is to write the sum of the series in a compact form using summation notation. Since the index of summation is denoted by \(i\), and the lower and upper limits of the series are 1 and 15 respectively, and the terms in these series is \(i^{2}\), it can be represented as \(\sum_{i=1}^{15}i^{2}\)

Key Concepts

Understanding Series Representation

When we talk about series representation, we usually refer to expressing a sequence of terms in a compact, mathematical way using summation notation. This enables us to work with long sums efficiently without listing out each term individually. For the series given in the exercise, the numbers from 1 to 15 are squared and added together. So, the series can be represented as:

• A compact form of writing repeated addition
• An expression that clearly shows the pattern involved
• A convenient notation to show start and end of the sequence

The summation notation uses the Greek letter sigma (\(\Sigma\)) to denote that a sum of terms is present. By defining the general term of the sequence as \(i^2\), we capture the essence of the series without needing to explicitly write each term. This form is especially useful in scenarios where the number of terms is large or mainly follows a specific pattern.

What is the Index of Summation?

The index of summation is the variable used to represent the position of a term within a sequence. In the given exercise, the variable \(i\) acts as the index of summation. It plays a central role in the expression of series as it defines:

• The term to be evaluated at each step of the summation.
• The variable that iterates through all positions from the lower to the upper limit.

In this specific exercise, \(i\) begins at 1 and increases by 1 as it takes on each successive integer value up to 15. It can be seen as a counter that helps generate the terms of the series, positioning itself at each integer within the defined range. Thus, the index of summation ensures that each term (\(i^2\)) is included in the series calculation, making it a fundamental component of the notation.

Exploring the Limits of Summation

The limits of summation define the boundaries of the series, indicating the first and last terms we are interested in summing. In summation notation, these are expressed as two numbers accompanying the index of summation below and above the sigma symbol.

• The lower limit defines where the summation begins.
• The upper limit marks where the summation ends.

For this exercise, with the representation \(\sum_{i=1}^{15}i^{2}\), the lower limit is 1, showing that the sum begins with the term \(1^2\). The upper limit is 15, indicating the series concludes with \(15^2\). These limits ensure we are calculating the sum for the correct set of terms, providing precision and clarity in the summation process. Choosing correct limits is vital as they directly affect the outcome of the series.