Problem 64 - Precalculus: Mathematics for Calculus

Question

Write the sum without using sigma notation. $$\sum_{k=6}^{9} k(k+3)$$

Step-by-Step Solution

Answer

The sum is 320.

Step 1: Identify the Terms

The given sum is \( \sum_{k=6}^{9} k(k+3) \). We need to identify each term in the sum by substituting the values from \( k = 6 \) to \( k = 9 \) into the expression \( k(k+3) \).

Step 2: Calculate Each Term

Calculate each term by substituting the value of \( k \):1. For \( k = 6 \), the term is \( 6(6+3) = 6 \times 9 = 54 \).2. For \( k = 7 \), the term is \( 7(7+3) = 7 \times 10 = 70 \).3. For \( k = 8 \), the term is \( 8(8+3) = 8 \times 11 = 88 \).4. For \( k = 9 \), the term is \( 9(9+3) = 9 \times 12 = 108 \).

Step 3: Write the Sum

Now, write the sum of the calculated terms:\[54 + 70 + 88 + 108\].

Step 4: Compute the Total Sum

Add all the terms together:\(54 + 70 + 88 + 108 = 320\).

Key Concepts

Discrete mathematics

Discrete mathematics is a fundamental area of mathematics that deals with distinct and separate values, rather than continuous ranges. Unlike calculus, which often involves smoothly varying quantities, discrete mathematics involves structures like integers, graphs, and logical statements. Here are key points about discrete mathematics:

• Study of Countable Entities: It deals with countable, distinct, and separate values, like whole numbers or graphs.
• Applications: Discrete math is crucial for computer science, cryptography, and network modeling, where structures are naturally discrete.
• Importance: It's essential for developing algorithms and understanding the underlying principles of digital structures.

In the context of the given exercise, discrete mathematics provides a framework for understanding mathematical concepts that are separated into distinct parts, such as evaluating the sum across a set of integers.

Sigma notation

Sigma notation is a powerful way to represent the sum of a sequence of terms. It uses the Greek letter sigma (\(\Sigma\)) and is an integral part of many areas in mathematics, including discrete math and calculus. Here's how it works:

• Structure: It involves an expression \(\Sigma_{k=m}^{n} a_k\), where \(m\) is the starting index, \(n\) is the ending index, and \(a_k\) is an expression involving the index \(k\).
• Benefits: It provides a compact notation to describe lengthy sums without writing out numerous terms explicitly.
• Usage: This notation is especially useful for expressing sums that follow a consistent or repetitive pattern.

In the exercise, sigma notation gives a concise way to represent the sum of \(k(k+3)\) for values from 6 to 9, offering a neat and generalized way to handle potentially long arithmetic operations.

Arithmetic sequences

Arithmetic sequences are lists of numbers with a constant difference between consecutive terms. This difference is known as the "common difference." Understanding these sequences is crucial for many mathematical problems like summation. Here's a detailed look:

• Definition: A sequence where each term after the first is found by adding a constant \(d\) (the common difference) to the previous term.
• Formula: The \(n\)th term of an arithmetic sequence can be calculated using \(a_n = a_1 + (n-1) imes d\), where \(a_1\) is the first term.
• Sum of Sequence: The sum of an arithmetic sequence can be found using the formula \((n/2) imes (a_1 + a_n)\)

In our exercise, we're actually working with a series of values for \(k\), which contribute to the arithmetic sequence through the expression \(k(k+3)\), forming the terms that are added together. Understanding arithmetic sequences helps in recognizing these patterns and simplifying sum calculations.