Problem 43 - College Algebra

Question

Express each sum using summation notation. Use 1 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 expression can be written in summation notation as \(\sum_{i=1}^{15} i^{2}\)

Step 1: Write Summation Notation

Summation notation, or sigma (\(\sigma\)) notation, is a concise way to represent the sum of multiple terms that follow a pattern. For this particular exercise, the sequence is of squares \(i^{2}\) from 1 to 15. To write this in sigma notation, note that the lower limit of the summation (written below the sigma) is 1, and the upper limit (written above the sigma) is 15. The index of summation is the variable \(i\). Thus the summation notation for \(1^{2}+2^{2}+3^{2}+\dots+15^{2}\) is: \(\sum_{i=1}^{15} i^{2}\)

Step 2: Identify the relevant trigonometric identities

Based on the given expression or equation, identify which trigonometric identities (Pythagorean, double-angle, sum/difference, etc.) are applicable.

Step 3: Apply the identities and simplify

Apply the identified identities to transform the expression. Simplify step by step, combining like terms and reducing fractions where possible.

Step 4: Solve or evaluate

If solving an equation, isolate the trigonometric function and find the angle(s). If evaluating, compute the final numerical value.

Step 5: State the result

Express the final answer, including all solutions in the required domain if solving an equation.

Step 6: Conclude with the answer

The expression can be written in summation notation as \(\sum_{i=1}^{15} i^{2}\)

Key Concepts

Sigma Notation

Sigma notation is a mathematical way to represent a series of terms that you want to sum together. This symbolic representation uses the Greek letter sigma (\( \Sigma \)) to denote the sum. Here's how it works:

• The starting point, or lower limit, is written below the sigma symbol.
• The ending point, or upper limit, is written above the sigma.
• The expression that describes each term in the series is written to the right of the sigma.

The benefit of sigma notation is its ability to succinctly capture long and repetitive addition sequences, allowing mathematicians and students to easily manage complex problems. For example, the sum of squares from 1 to 15 can be represented in sigma notation as: \(\sum_{i=1}^{15} i^{2} \). This notation tells you to evaluate the expression \(i^2\) for each integer \(i\) starting at 1 and ending at 15, then add up all those values.

Sequence

A sequence in mathematics is an ordered list of numbers. Each number in the list is called a term. Sequences often follow specific rules or patterns, making them predictable. In the original problem, the sequence consists of the squares of the first 15 positive integers: 1, 4, 9, and so on, up to 225.
The key characteristics of a sequence include:

• A defined starting point, often called the first term.
• A rule that describes how to get from one term to the next.
• A potentially infinite continuation, but a sequence can also be finite.

Recognizing the pattern of a sequence allows you to easily model and calculate its terms. In our case, the pattern is squaring each successive integer: \(n^2\), where \(n\) is the term number in the sequence, starting from 1.

Index of Summation

The index of summation is the variable in sigma notation that tells you which term you're currently working on in the sum. In our context, the index of summation is \(i\). This index appears in the expression next to the sigma symbol, guiding the evaluation of each term in the series.
For the sum written as \(\sum_{i=1}^{15} i^{2}\), the index \(i\) will take on several integer values, starting from the lower limit (1) and ending at the upper limit (15).
The index serves a few important roles:

• It sequentially tracks each term of the summation.
• It indicates the position of the current term.
• It provides a counter in cases where the pattern involves more complex calculations.

Understanding the use of the index is necessary for evaluating each term and ensuring the entire sum is calculated correctly.

Squares

Squaring a number means multiplying that number by itself. For example, squaring 3 gives you 9, because \(3 \times 3 = 9\). In the context of a sequence or summation problem, dealing with squares often means recognizing a recurring pattern.
Squares grow quickly as numbers increase:

• The square of a small number is also small, like \(2^2 = 4\).
• The square of a larger number increases rapidly, such as \(10^2 = 100\).

The exercise you looked at involves summing the squares of numbers from 1 to 15. This is a good example of how a simple operation—squaring—can appear in a more complex mathematical form, such as a series sum.
Being able to recognize and calculate squares is a crucial skill not only in sequence problems but in various other mathematical contexts as well.