Problem 15 - Algebra 2

Question

Use summation notation to write each arithmetic series for the specified number of terms. $$ 5+6+7+\ldots ; n=7 $$

Step-by-Step Solution

Answer

The arithmetic series \(5+6+7+... ; n=7\) is written in summation notation as \(\sum_{i=1}^{7}(i + 4)\)

Step 1: Understand the arithmetic series

In an arithmetic series, each term after the first is obtained by adding a constant difference to the preceding term. In this series, the difference between each term is 1.

Step 2: Write the general term

The nth term of an arithmetic series can be written as \(a + (n-1)d\), where \(a\) is the first term, \(n\) is the term number and \(d\) is the common difference. In this series, \(a = 5\), \(d = 1\). So the nth term \(T_n\) is \(5 + (n-1)1 = 5 + n -1 = n + 4\).

Step 3: Write in a summation notation

With the general term defined, the series can be written in summation notation, \(\sum_{i=1}^{n}(i + 4)\), where \(i\) is starting from 1 to \(n\) (n is 7 in this case).

Key Concepts

Arithmetic Series

An arithmetic series is a type of sequence where the terms are obtained by adding a constant value, known as the common difference, to the previous term. This series represents the sum of the elements of an arithmetic sequence.
When working with arithmetic series, you often need to find the sum of all terms up to a certain point. This can be efficiently represented using summation notation, which uses the symbol \( \Sigma \). In this exercise, the arithmetic series given is \(5 + 6 + 7 + \ldots\) and we denote it using summation as \( \sum_{i=1}^{7}(i + 4) \).
Understanding the transition from an arithmetic sequence to an arithmetic series is crucial, as it allows us to succinctly summarize sequences and find their sums in a compact form.

Common Difference

In an arithmetic sequence, the common difference is the amount you consistently add to each term to get the next term. This difference is what defines the structure and progression of the sequence.
In the context of the provided series \(5, 6, 7, \ldots\), the common difference is 1. We recognize this because each term increases by 1 from the previous term:

• From 5 to 6, we add 1.
• From 6 to 7, we add 1.

Identifying the common difference is essential for calculating subsequent terms and for correctly expressing the series using summation notation. Knowing this value allows us to develop a formula for any term in the sequence.

General Term

The general term of an arithmetic sequence gives you a formula to find any term in the sequence without listing all the previous ones. This means you can find, say, the 20th term with just a little math, rather than listing each term individually.
The general term is written as \( a + (n-1)d \), where \( a \) is the first term of the sequence, \( n \) is the term number, and \( d \) is the common difference. For our sequence beginning at 5 with a common difference of 1, the general term is \( n + 4 \).
This formula is key when transitioning from a list of sequence terms to a summation of a series, letting you express any term within the series precisely.

Arithmetic Sequence

An arithmetic sequence is a list of numbers where each number is derived by adding a constant value, known as the common difference, to the previous number. This pattern of adding makes the sequence predictable and easy to analyze.
In the given exercise, the sequence starts at 5 and proceeds with a common difference of 1:

• 5 is the first term.
• 6 is derived by adding 1 to the first term.
• 7 is the result of adding 1 to 6, and so forth.

By understanding this sequence, students can easily determine any subsequent terms without extensive calculation, facilitating the process of forming the arithmetic series from it. A firm grasp of arithmetic sequences also aids in effectively using summation notation to express these patterns neatly and concisely.