Problem 32 - College Algebra

Question

Find each indicated sum. $$\sum_{-1}^{5} i^{3}$$

Step-by-Step Solution

Answer

The sum of the cubes of integers from -1 to 5 is 224.

Step 1: Identify the Range

The range of numbers to be cubed is from -1 to 5. So the set of numbers to use is \{-1, 0, 1, 2, 3, 4, 5\}.

Step 2: Apply the Function

The function to apply is cubing. Applying cube to each number: \[(-1)^3, (0)^3, (1)^3, (2)^3, (3)^3, (4)^3, (5)^3\] yields \{-1, 0, 1, 8, 27, 64, 125\} respectively.

Step 3: Sum the Results

Add all the calculated values together: \[ -1 + 0 + 1 + 8 + 27 + 64 + 125 = 224 \]

Key Concepts

Cube function

The cube function is a mathematical operation where a number is multiplied by itself twice, resulting in what's called a "cubed" number. This process is denoted as raising a number to the power of three. For example, if you cube 2, you perform the operation \( 2^3 = 2 \times 2 \times 2 = 8 \). This operation is quite straightforward.

• Start with a number: this could be any integer or real number.
• Multiply the number by itself. This means you do it twice: \( x \times x \times x \).
• Get the result, known as the cube of the original number.

Using the cube function can reveal volume relationships in geometry, as cubes are three-dimensional objects, and cubing relates to the calculation of the volume of such objects. Understanding the cube function is essential when dealing with series and various algebraic expressions involving powers.

Arithmetic series

An arithmetic series is a sequence of numbers in which each term after the first is produced by adding a fixed, constant number known as the common difference. This type of series is expressed in the sum of its terms. For example, the series \( 3, 7, 11, 15 \) is arithmetic because each term is 4 more than the previous one.To find the sum of an arithmetic series, you can use the formula: \[S_n = \frac{n}{2} \times (a + l)\]where:

• \( S_n \) is the sum of the first \( n \) terms,
• \( n \) is the number of terms,
• \( a \) is the first term,
• \( l \) is the last term.

In our specific case, we were dealing with a sum of cubes from \( i = -1 \) to \( i = 5 \), which is a sequential process involving each integer in the range rather than a classic arithmetic sequence. However, understanding the role of a series helps in grasping the processes behind summation and accumulations.

Algebraic expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They represent mathematical relationships and can be used to simplify calculations and solve equations. For example: \( 3x^2 + 2x + 1 \) is an algebraic expression where numbers multiply by variables or raised to powers.Key components of algebraic expressions:

• **Constants**: Numbers without variables, like 1 or 5.
• **Variables**: Symbols usually represented by letters like \( x \), \( y \), corresponding to numbers.
• **Coefficients**: Numbers multiplying variables, such as 3 in \( 3x \).
• **Operators**: Addition, subtraction, multiplication, division signs that combine or modify elements.
• **Exponents**: Show repeated multiplication, as seen with cubing, such as \( x^3 \).

Algebraic expressions can model real-world phenomena and enable complex problem solving, allowing us to find unknown values and relationships in equations. Understanding them is fundamental to tackling equations involving cube functions and series.