Question

Write an expression in sigma notation for each situation. The sum of the cubes of the first 5 odd integers.

Short answer

Question: Find a sigma notation expression to represent the sum of the cubes of the first 5 odd integers. Answer: \(\sum_{n=1}^{5} (8n^3 - 12n^2 + 6n - 1)\)

Step-by-step solution

Identify the general term for odd integers

The general term for odd integers can be represented as 2n - 1, where n is a positive integer.

Cube the general term

Now, we need to cube the expression (2n - 1). To do this, we can use the formula (a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3. So, cubing (2n - 1) would be: (2n - 1)^3 = (2n)^3 - 3(2n)^2(1) + 3(2n)(1)^2 - (1)^3

Simplify the cubed expression

Now, let's simplify the cubed expression: (2n - 1)^3 = 8n^3 - 12n^2 + 6n - 1

Write the sigma notation expression

Finally, we need to sum the cubes of the first 5 odd integers. We do this by setting the lower limit as n = 1 and the upper limit as n = 5 to the sigma notation. So, the sigma notation expression for this situation is: \(\sum_{n=1}^{5} (8n^3 - 12n^2 + 6n - 1)\)

Key concepts

Odd Integers

Odd integers are numbers that cannot be divided evenly by 2. They have remainders when divided by 2, which gives them a unique placement in the number line. A common pattern to detect odd integers is their alternating nature, where they appear in sequences like 1, 3, 5, 7, and so on. To find a general representation of odd integers, we often use the formula \(2n - 1\), where \(n\) is a positive integer. This formula helps in generating any odd integer by substituting \(n\) with sequential natural numbers.

• Example: For \(n = 1\), \(2n - 1 = 1\), giving the first odd integer.
• For \(n = 2\), \(2n - 1 = 3\), giving the second odd integer.
• This method continues to accurately list the odd numbers as you increment \(n\).

Understanding this concept is vital for calculating expressions and creating sequences of these numbers, including their sums, products, or other operations.

Cubes of Integers

Cubing an integer means multiplying the integer by itself twice. In mathematical terms, the cube of any number \(x\) is expressed as \(x^3\). When calculating the cube of an integer, you are determining the volume of a cube with side length \(x\). Cubes grow quite rapidly as integers increase, illustrating exponential growth. For example:

• The cube of 2, \(2^3\), is 8.
• The cube of 3, \(3^3\), becomes 27.
• Interestingly, cubes of numbers play a significant role in geometry and algebraic expressions.

When we are interested in the cubes of odd integers, such as in the given exercise, the algebraic representation \((2n - 1)^3\) is used. We expand this expression using the binomial theorem or other algebraic identities, like: \[ (a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3 \] This identity simplifies our computation, helping to derive cubes of odd integers efficiently.

Sum of Series

The sum of a series is a concept of adding up a sequence of terms. In mathematics, sigma notation \(\sum\) is used to efficiently express this operation. Sigma notation allows for the clear and concise writing of long sums in terms of a pattern or general rule. In the context of the exercise, we are tasked with finding the sum of the cubes of the first five odd integers. Using sigma notation, you can express the sum of terms generated by \((2n - 1)^3\) from \(n = 1\) to \(n = 5\). This is written as:\[ \sum_{n=1}^{5} (8n^3 - 12n^2 + 6n - 1) \]This formula guides you in adding each term from \(n = 1\) to \(n = 5\) by simply substituting \(n\) in the expression and summing up the results. The use of sigma notation simplifies the expression of large sums and makes it easier to identify patterns or properties of the sequence involved. By understanding the straightforward assembly and use of sigma notation, tackling problems involving series and sequences becomes much more manageable, making computations clearer and often more efficient.