Problem 1 - Algebra 2

Question

Write the related series for each finite sequence. Then evaluate each series. $$ 21,18,15,12,9,6,3 $$

Step-by-Step Solution

Answer

The sum \(S\) of the arithmetic series associated with the finite sequence is 168.

Step 1: Identify the first term, last term and common difference

Looking at the sequence, it can be observed that the sequence decreases by 3 each time, thus the common difference \(d\) is -3, the first term \(a_1\) is 21, and the last term \(a_7\) is 3.

Step 2: Find the number of terms

Count the terms in the sequence to find out the number of terms \(n\). In this case, \(n = 7\).

Step 3: Use the arithmetic series formula

Substitute \(n\), \(a_1\), and \(a_n\) into the arithmetic series formula \(\frac{n}{2}(a_1 + a_n)\) to evaluate the series. Substituting the values we have, gives: \(\frac{7}{2}(21 + 3) = \frac{7}{2} * 24 = 14 * 12 = 168\). Therefore, the sum \(S\) of this arithmetic series is 168.

Key Concepts

Finite Sequence

A finite sequence is a list of numbers that has a defined number of elements. In simple terms, it means that the sequence does not go on forever—it has both a starting point and an ending point.
For example, the sequence 21, 18, 15, 12, 9, 6, 3 has exactly seven numbers. Once you reach the last number, the sequence comes to a stop. This is different from an infinite sequence, which would continue indefinitely without stopping.
Understanding that this sequence is finite is crucial when working with arithmetic series, because it allows you to calculate the total sum using all the numbers listed from start to finish.

• Start and End: You can clearly see the first and last number.
• Countable: You can count how many numbers are in the sequence without any ambiguity.

Common Difference

In an arithmetic sequence, the common difference is the consistent interval between one number and the next.
In the given sequence 21, 18, 15, 12, 9, 6, 3, the common difference can be found by subtracting any term from the term that follows it.
For this series:

• 18 - 21 = -3
• 15 - 18 = -3

This sequence consistently decreases by 3, hence the common difference is -3.
The idea of the common difference is what allows you to verify that a sequence is indeed arithmetic—because this difference remains constant.
Knowing the common difference helps you to predict subsequent numbers of the sequence or check its regularity.

First Term

The first term of a sequence is simply the very first number or element in it. In this arithmetic sequence, the first term is 21.
The importance of the first term cannot be overstated because it is necessary for many calculations you might perform, such as finding the series' sum or predicting future values in the pattern.
When you use formulas related to arithmetic series, like the one for finding the total sum, the first term (\(a_1\)) is always required.

• This initial value is your starting point when looking at the series as a whole.

It sets the stage to determine each subsequent term given the common difference.

Number of Terms

The number of terms in a sequence tells you exactly how many numbers are included from start to end. In the linear pattern 21, 18, 15, 12, 9, 6, 3, there are seven terms.
This number,\(n\),is fundamental for applying formulas and understanding the full scope of the series.
It is important because:

• It allows you to calculate the sum using the arithmetic series formula.
• It indicates how many intervals exist between the start and the stop of the sequence.

In arithmetic sequences, knowing \(n\)gives insight into the complete structure of the sequence, allowing you to comprehend its full range.