Problem 12 - Calculus Single Variable

Question

In Exercises \(11-16,\) use summation notation to express the sum. 3+6+9+12+15

Step-by-Step Solution

Answer

The sum is expressed as \( \sum_{n=1}^{5} 3n \).

Step 1: Identify the Pattern

Observe the sequence: 3, 6, 9, 12, 15. Notice that each term increases by 3 compared to the previous one.

Step 2: Define the Sequence Formula

Identify the first term of the sequence as 3. The common difference is 3. The sequence can be expressed in terms of its position: the term can be represented as \(3n\), where \(n\) is the position of the term in the sequence (starting from 1).

Step 3: Determine the Range of the Sequence

The sequence starts with 3 at \(n=1\) and continues to 15. Determine the number of terms by checking: \(3n = 15\). Solving for \(n\), we get: \(n = 5\). Thus, the range of \(n\) is from 1 to 5.

Step 4: Write the Sum in Summation Notation

Using the sequence formula \(3n\) and the range from \(n=1\) to \(n=5\), the sum can be written as: \[ \sum_{n=1}^{5} 3n \]

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is constant. This constant difference is known as the "common difference." Let's consider the example of our sequence: 3, 6, 9, 12, 15. Here, if you look closely at the sequence, you'll notice that each term increases by a fixed number, which is 3. This fixed number is what we refer to as the common difference of the arithmetic sequence.
The constant addition of this common difference is what grows the sequence. By understanding this pattern, predicting any future term in the sequence becomes straightforward. If you want the next number after 15, you just add the common difference, which would give you 18.

Sequence Formula

The sequence formula provides a method to find any term in an arithmetic sequence without listing all preceding ones. It's like a shortcut! To derive a sequence formula, you need the first term of the sequence and the common difference. Let's say the first term in a sequence is denoted by "a." The formula to find the nth term in an arithmetic sequence is given by:
\[ a_n = a + (n-1) imes d \]
Where:

• \(a_n\) = nth term of the sequence
• \(a\) = first term of the sequence
• \(d\) = common difference
• \(n\) = position of the term in the sequence

For our example sequence (3, 6, 9, ...), the first term \(a\) is 3, and the common difference \(d\) is also 3. Therefore, the sequence formula becomes:
\[ a_n = 3 + (n-1) imes 3 \]
This simplifies to \(a_n = 3n\), showing a direct relationship between the term's position and its value.

Common Difference

The common difference in an arithmetic sequence is the amount that each term increases by. This concept is crucial for both understanding the sequence and using its formula.
In our example sequence, each number increases by 3. This is the constant increment that each term adds to the previous one.
Understanding the common difference helps us explore several things:

• How quickly the sequence is growing
• Predicting future terms
• Recognizing arithmetic sequences among sets of numbers

To see how the common difference is calculated, you simply subtract any term in the sequence from the term that follows it. Using the sequence example, calculating the common difference goes as follows: \(6 - 3 = 3, 9 - 6 = 3\), and so on. Each subtraction confirms the arithmetic nature of the sequence with a common difference of 3.