Problem 13 - Calculus Single Variable

Question

In Exercises \(11-16,\) use summation notation to express the sum. $$ 9+13+17+21+25+29 $$

Step-by-Step Solution

Answer

The sum can be expressed as \( \sum_{i=1}^{6} (9 + (i-1)4) \).

Step 1: Identify the Sequence

Observe the sequence of numbers presented in the problem: 9, 13, 17, 21, 25, and 29. Notice that this sequence is formed by increasing numbers where each successive number increases by 4.

Step 2: Determine the Common Difference

The common difference, denoted as \( d \), is found by subtracting any two consecutive terms in the sequence. Here, \( 13 - 9 = 4 \), confirming that \( d = 4 \).

Step 3: Determine the First Term

Identify the first term of the sequence, which we denote as \( a_1 \). For this problem, \( a_1 = 9 \).

Step 4: Establish the General Term Formula

The formula to find any term in an arithmetic sequence is \( a_n = a_1 + (n-1) \, d \). Plug in the first term and the common difference: \( a_n = 9 + (n-1) \, 4 \).

Step 5: Express the Sequence in Summation Notation

To express the sequence as a summation from 1 through 6, use the formula for each term derived in Step 4:\[ S = \sum_{i=1}^{6} (9 + (i-1) \, 4) \]

Step 6: Confirm the Number of Terms

There are 6 terms in the series: 9, 13, 17, 21, 25, 29. Thus, \( n = 6 \), which corresponds with the upper limit of the summation notation.

Key Concepts

Summation Notation

Summation notation is a concise way of expressing the sum of a series of numbers, particularly useful for arithmetic sequences. In our example, let's focus on the arithmetic sequence 9, 13, 17, 21, 25, and 29.
To express this in summation notation, we use the symbol \( \Sigma\) (sigma), which denotes the sum, followed by an index that represents each term. It typically looks like this:

• \( S = \sum_{i=1}^{n} a_i \)


• Here, \( n \) is the number of terms, and \( a_i \) is the general term for each individual term in the sequence.


• For the sequence provided, we calculate it using: \( S = \sum_{i=1}^{6} (9 + (i-1) \cdot 4) \).

This notation efficiently encapsulates the entire series in one compact expression, making it easier to work with larger sequences.

Common Difference

The common difference is a key feature of arithmetic sequences. In simple terms, it's the amount by which the sequence increases or decreases with each successive term.
For the sequence 9, 13, 17, 21, 25, 29, the common difference, \( d \), can be found by subtracting any two consecutive terms:

• Take the second term and subtract the first term: \( 13 - 9 = 4 \).


• Choose any other consecutive terms, like \( 17 - 13 \), and you will also find \( d = 4 \).


• This repetitiveness confirms uniformity in the progression of the sequence, essential in arithmetic sequences.

Recognizing the common difference helps understand how the sequence progresses and provides a foundation for deriving formulas.

General Term Formula

The general term formula allows you to find any term in an arithmetic sequence without listing all the preceding terms. It provides a way to calculate \( a_n \), the \( n \)-th term of the sequence, given the first term and the common difference.

You'd use the following formula:

• \( a_n = a_1 + (n-1) \cdot d \)


• Here, \( a_1 \) is the first term, \( n \) is the term number you're looking for, and \( d \) is the common difference.


• For our sequence, \( a_n = 9 + (n-1) \cdot 4 \).

This formula simplifies calculating any term, saving time and effort, especially in long sequences. Understanding its derivation and application is crucial for solving arithmetic sequence problems.

Series

A series is the expression of the sum of the terms of a sequence. When dealing with arithmetic sequences, you often want to find the sum of a certain number of terms, referred to as an arithmetic series.

In our example, we explored the sequence 9, 13, 17, 21, 25, 29. Turning this sequence into a series involves using summation notation:

• \( S = \sum_{i=1}^{6} (9 + (i-1) \cdot 4) \)


• This represents adding the sequence's terms for \( i \) from 1 to 6.

The idea of a series not only serves mathematical curiosity but also provides valuable insight into patterns, particularly when handling larger datasets. With the arithmetic series formula, you can quickly determine the accumulated total value of a sequence, which is highly practical in various fields, such as finance and computer science.