Question

Write out the sums. (You do not need to evaluate them.) \(\sum_{i=0}^{5} i^{2}\)

Short answer

Question: Calculate the value of the sum \(\sum_{i=0}^{5} i^{2}\). Answer: To calculate the value of the sum \(\sum_{i=0}^{5} i^{2}\), we need to substitute the index variable i with its values in the given range, from 0 to 5, and add the corresponding terms: \(\sum_{i=0}^{5} i^{2} = (0)^{2} + (1)^{2} + (2)^{2} + (3)^{2} + (4)^{2} + (5)^{2} = 0 + 1 + 4 + 9 + 16 + 25 = 55\) Thus, the value of the sum is 55.

Step-by-step solution

Understand the exercise

We have a sum with the index variable i, which starts from 0 and goes to 5, of the term \(i^2\).

List the terms of the sum

We need to find the terms of the sum by substituting the index variable i with its values in the given range, from 0 to 5, and write them out: For i = 0: \((0)^{2}\) For i = 1: \((1)^{2}\) For i = 2: \((2)^{2}\) For i = 3: \((3)^{2}\) For i = 4: \((4)^{2}\) For i = 5: \((5)^{2}\)

Write out the sum

Now we write all these terms with the addition sign between them as the sum: \(\sum_{i=0}^{5} i^{2} = (0)^{2} + (1)^{2} + (2)^{2} + (3)^{2} + (4)^{2} + (5)^{2}\)

Key concepts

Index Variable

In summation, the **index variable** serves as a crucial component of mathematical notation used to specify the terms within the sum. The index variable is essentially a placeholder that represents each integer in a specified range. For example, in the summation notation \(\sum_{i=0}^{5} i^{2}\), the letter \(i\) acts as the index variable. Typically, the index variable is shown with the range it covers below the summation symbol. In this exercise, \(i\) takes values from 0 to 5.

As you process each term in the sum, you replace the index variable with each integer in its range. For instance, \(i=0\), \(i=1\), \(i=2\), and so on, up to \(i=5\). By systematically substituting the index variable with these values, you write out distinct expressions—each an essential building block of the total sum. This methodical approach makes the concept of summation clear and organized.

Exponents

**Exponents** are a powerful tool in mathematics that denote repeated multiplication of a number by itself. The small superscript number you see is the exponent, which indicates how many times the base number should be used as a factor. In our exercise, each component of the sum is represented in terms of exponents with \(i^2\). Here, \(i\) is the base, and 2 is the exponent.

When we calculate the expression \(i^2\), it means multiplying the base \(i\) by itself one time, or \(i \times i\). For example:

• \((0)^{2} = 0 \times 0 = 0\)
• \((1)^{2} = 1 \times 1 = 1\)
• \((2)^{2} = 2 \times 2 = 4\)
• \((3)^{2} = 3 \times 3 = 9\)
• \((4)^{2} = 4 \times 4 = 16\)
• \((5)^{2} = 5 \times 5 = 25\)

Exponents simplify and efficiently express repeated calculations, helping make complex problems more manageable.

Mathematical Notation

**Mathematical notation** is a systematic method of expressing mathematical concepts in a concise and standardized way. It employs symbols and operators to make mathematics more understandable and precise. The notation \(\sum_{i=0}^{5} i^{2}\) is a perfect illustration of this.

Here's how to interpret this notation step by step:

• The sigma symbol \(\sum\) is used to denote the sum of a series.
• The expression \(i=0\) below \(\sum\) shows the starting value of the index variable.
• The number above the \(\sum\), in this case, 5, signifies the ending value for the index variable.
• \(i^{2}\) is the general term to be calculated at each step in the range specified by the index.

Understanding and using this notation allows us to convey complex mathematical ideas efficiently by condensing lengthy expressions into a structured form. It becomes a universal language that facilitates the communication of ideas across diverse audiences. By mastering mathematical notation, learners improve their capacity to solve and understand a wide range of problems with clarity and efficiency.