Problem 41 - Algebra 2

Question

Use summation notation to write each arithmetic series for the specified number of terms. $$ 41+33+25+\ldots ; n=8 $$

Step-by-Step Solution

Answer

The arithmetic series 41, 33, 25,...for n=8 terms can be written in summation notation as \(\sum_{i=1}^{8} [41 + (i-1)(-8)]\)

Step 1: Determine the first term (a) and the common difference (d)

In an arithmetic series, the first term and the common difference can be directly observed. In this case, the first term (a) is 41 and the common difference (d), obtained by subtracting the second term from the first term, is -8.

Step 2: Write the general formula for the nth term

In an arithmetic sequence, the nth term (a_n) can be given by the formula \(a_n = a + (n-1)d\). For this exercise, it translates into \(a_n = 41 + (n-1)(-8)\).

Step 3: Write the arithmetic series in summation notation

The given arithmetic series: 41, 33, 25,...for n=8 terms can be written in summation notation using the formula for the nth term. This series can be represented as \(\sum_{i=1}^{8} [41 + (i-1)(-8)]\).

Key Concepts

Summation Notation

Summation notation is a powerful mathematical tool that allows you to express the addition of a sequence of numbers in a compact format. It is represented using the Greek letter Sigma (\(\sum\)). This notation helps in simplifying the expression of long series, especially when they follow a specific pattern, such as arithmetic sequences. In the context of arithmetic series, summation notation is used to add a series of numbers where each term is derived from the nth term formula. For our example, the arithmetic series \( 41 + 33 + 25 + \ldots \) with 8 terms can be neatly written as \( \sum_{i=1}^{8} [41 + (i-1)(-8)] \). Here, the index \(i\) starts at 1 and ends at 8 to represent each term in the series.

nth Term Formula

The nth term formula is essential in identifying the specific terms in an arithmetic sequence or series. It gives you a rule to find any term in the sequence without writing out all the preceding terms. The formula is given by:

• \( a_n = a + (n-1)d \)

where:

• \( a_n \) is the nth term,
• \( a \) is the first term,
• \( n \) is the term number,
• \( d \) is the common difference.

Using this formula, you can determine any term of the sequence. In the example, the nth term is \( 41 + (n-1)(-8) \), helping us find each term in the series for the specified value of \(n\). This formula essentially builds the arithmetic sequence, one term at a time.

Common Difference

The common difference in an arithmetic sequence is the consistent difference between consecutive terms. It plays a crucial role in the calculation of each subsequent term. The formula to find the common difference \(d\) is:

• \( d = a_{n+1} - a_n \)

This simply means subtracting any term from the following term. In our given sequence, by subtracting the second term 33 from the first term 41, we derive the common difference \(d = 33 - 41 = -8\). This negative sign indicates that each subsequent term in the series decreases by 8.

First Term

The first term of an arithmetic sequence is the starting point from which the sequence develops. It is denoted by \(a\) and is a fundamental element in defining the sequence. In our example, the arithmetic series starts at 41, making it the first term \(a=41\). The first term, combined with the common difference, helps build the entire sequence using the nth term formula. Understanding the first term is pivotal because it establishes the initial condition of the series and influences all future terms generated by adding the common difference to it iteratively.