Problem 89 - Precalculus with Limits

Question

In Exercises 85-96, find the sum. \( \displaystyle \sum_{i=0}^{4} i^2 \)

Step-by-Step Solution

Answer

The required sum is 30.

Step 1: Understand the sum notation

The given expression, \( \displaystyle \sum_{i=0}^{4} i^2 \), means that we need to find the sum of the squares of all integers from 0 to 4. This is expressed in the form of a definite series, where the variable 'i' is the index of summation.

Step 2: Apply the sum formula

The sum of squares of the first n numbers is given by the formula \( \frac{n(n + 1)(2n + 1)}{6} \). However since here, we are summing from 0 instead of 1, we need to subtract the zeroth term (which is 0) from the sum upto 4.

Step 3: Calculation

Substituting the value n = 4 into the formula, the sum is \( \frac{4 \cdot 5 \cdot 9}{6} \). Using simple arithmetic, this simplifies to 30.

Key Concepts

Sum of Squares

The concept of the sum of squares refers to adding together the squares of a set of numbers. In mathematical terms, this is often expressed using summation notation as seen with \( \displaystyle \sum_{i=0}^{4} i^2 \). The expression requires you to square each number individually and add those squared values.

In our example, we have integers from 0 to 4. Here are their squares:

• 0 squared is 0
• 1 squared is 1
• 2 squared is 4
• 3 squared is 9
• 4 squared is 16

After squaring, we sum these values: 0, 1, 4, 9, and 16. The sum of these squares is 30.

Understanding the sum of squares is useful, especially in statistics and data analysis, where variations from a mean value often are expressed this way.

Index of Summation

The index of summation is a critical component when dealing with sum notation, represented usually by a variable within the summation symbol. In our expression \( \sum_{i=0}^{4} i^2 \), the 'i' is the index of summation.

This index tells you which values to consider in the series and ensures each is counted once. In our example, 'i' runs from 0 to 4. Hence:

• The first value is found by letting \( i = 0 \)
• The second value by letting \( i = 1 \)
• Continuing this way until \( i = 4 \)

The control provided by the index of summation allows us to systematically apply operations — in this case, squaring — to each integer and sum them efficiently. Through this index, the expression clearly delineates the bounds of calculation.

Definite Series

A definite series refers to the sum of terms defined by a particular expression, where the series has specific lower and upper limits. This is different from an indefinite series, which continues without a terminal point.

In \( \sum_{i=0}^{4} i^2 \), we see a definite series because the index 'i' takes on values in a fixed range, from 0 to 4 inclusive.

• The lower limit is 0, meaning 'i' starts at 0
• The upper limit is 4, meaning 'i' ends at 4

Such series provide the advantage of clear boundaries, making them perfect for calculations that require finiteness. Understanding definite series helps methodically resolve many mathematical problems where comprehending the bounds of calculation is essential.