Problem 55 - Beginning and Intermediate Algebra

Question

Write each series using summation notation. 3+6+9+12

Step-by-Step Solution

Answer

The arithmetic series 3+6+9+12 can be represented using summation notation as \(\displaystyle\sum\limits_{k=1}^4 (3 + (k-1)*3)\).

Step 1: Identify the first term, common difference, and number of terms in the series

The arithmetic series is 3+6+9+12. The first term, denoted as a_1, is 3. The common difference, which is the difference between any two consecutive terms, is 6 - 3 = 3. The number of terms, denoted as n, is 4 as there are four numbers in the series.

Step 2: Write the general term of the arithmetic series

To find the general term of an arithmetic series, we can use the formula: a_n = a_1 + (n-1)d where 'a_n' is the nth term, 'a_1' is the first term, 'd' is the common difference, and 'n' is the term number. In our case, the formula becomes: a_n = 3 + (n-1) * 3

Step 3: Write the series using summation notation

Now that we have the general term, we can write the series using summation notation. The summation notation, denoted as \(\displaystyle\sum\limits_{k=1}^n a_k\), represents the sum of a sequence using a specific formula for each term. In our case, the series can be represented as: \(\displaystyle\sum\limits_{k=1}^4 (3 + (k-1)*3)\) This notation represents the sum of the arithmetic series 3+6+9+12.

Key Concepts

Arithmetic Series

An arithmetic series is a series of numbers in which the difference between any two successive numbers, known as the 'common difference', is constant.

• It starts with a first term, often denoted as \(a_1\).
• The sequence is generated by adding the common difference to the previous term repeatedly.

In our given exercise, the series provided is 3 + 6 + 9 + 12. Here, the first term is 3. Building upon this, we can see that 6 is obtained by adding 3, and similarly, 9 is obtained by adding 3 to 6, thereby creating a pattern. Through this consistent addition, the series extends, maintaining its arithmetic nature.
Understanding the structure of arithmetic series is foundational as it is regularly implemented in various mathematical and real-life applications. This knowledge supports solving problems involving sequence predictions and summations.

Common Difference

The 'common difference' in an arithmetic series is the amount added to each term to reach the next term. It is denoted by the symbol \(d\), and calculated by subtracting any term from the following term.

• Given two consecutive terms \(a_2\) and \(a_1\), the common difference is \(d = a_2 - a_1\).

In our exercise, for the series 3 + 6 + 9 + 12, we can calculate the common difference by subtracting 3 from 6, yielding 3. This pattern holds for each pair of consecutive numbers in the series, further demonstrating that the series is arithmetic.
Comprehending the common difference is crucial as it aids in predicting further terms in the series and is instrumental when determining the general term expression of the series.

General Term Formula

The general term formula of an arithmetic series allows us to find any term within the sequence without having to extend it manually. The formula is:\[a_n = a_1 + (n-1) \cdot d\]Here:

• \(a_n\) represents the nth term we want to find.
• \(a_1\) is the first term of the series.
• \(d\) is the common difference.

Using our exercise example, the formula becomes \(a_n = 3 + (n-1) \cdot 3\). This formula matches the provided series by substituting values systematically; determining each term directly.
The power of the general term formula lies in its ability to calculate any desired term quickly, reinforcing the efficiency and elegance of mathematical theory in operations and real-life applications.

Summation Notation Formula

Summation notation is a compact mathematical representation of adding up terms in a sequence. It's symbolized by the Greek letter \(\Sigma\) and is useful for expressing the sum of sequences like arithmetic series. The notation reads:\[\sum_{k=1}^n a_k\]Here:

• \(k\) is the index of summation, starting from 1 in this case.
• \(n\) is the number of terms to add.

For our exercise, the series 3 + 6 + 9 + 12 can be expressed as:\[\sum_{k=1}^4 (3 + (k-1)\cdot 3)\]This translates the entire series into a single, concise mathematical expression, highlighting the elegance and power of summation notation for complex sequences and general mathematical operations. Understanding this notation is fundamental for dealing with sequences in higher levels of mathematics.