Question

In Exercises 2-4, complete the tables with the terms of the arithmetic series \(a_{1}, a_{2}, \ldots, a_{n},\) and the sequence of partial sums, \(S_{1}, S_{2}, \ldots, S_{n} .\) State the values of \(a_{1}\) and \(d\) where \(a_{n}=a_{1}+(n-1) d\) The table for the terms of the series \(2,6,10, \ldots,\) is $$ \begin{array}{l|l|l|l|l|l|l|l|l} \hline n & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline a_{n} & 2 & 6 & 10 & & & & & \\ \hline S_{n} & 2 & 8 & & & & & & \\ \hline \end{array} $$

Short answer

Answer: The partial sum of the first 8 terms of the series is 128.

Step-by-step solution

Identify \(a_1\) and \(d\)

The arithmetic sequence is given: \(2, 6, 10, \ldots\). From this, we can see that the first term is \(a_1 = 2\). The common difference, \(d\), is the constant difference between consecutive terms. Therefore, \(d = 6 - 2 = 4\).

Find the remaining terms of the series

Using the formula \(a_n = a_1 + (n-1)d\), we can find the other terms of the series: 1. For \(n=5\), \(a_5 = 2 + (5-1)4 = 2 + 16 = 18\) 2. For \(n=6\), \(a_6 = 2 + (6-1)4 = 2 + 20 = 22\) 3. For \(n=7\), \(a_7 = 2 + (7-1)4 = 2 + 24 = 26\) 4. For \(n=8\), \(a_8 = 2 + (8-1)4 = 2 + 28 = 30\)

Calculate the partial sums

To compute the partial sums, we simply add the terms of the series up to the given index: 1. \(S_4 = a_1 + a_2 + a_3 + a_4 = 2 + 6 + 10 + 14 = 32\) 2. \(S_5 = S_4 + a_5 = 32 + 18 = 50\) 3. \(S_6 = S_5 + a_6 = 50 + 22 = 72\) 4. \(S_7 = S_6 + a_7 = 72 + 26 = 98\) 5. \(S_8 = S_7 + a_8 = 98 + 30 = 128\)

Fill in the table

Now that we have found the missing terms and partial sums, we can complete the table: $$ \begin{array}{l|l|l|l|l|l|l|l|l} \hline n & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\\ \hline a_{n} & 2 & 6 & 10 & 14 & 18 & 22 & 26 & 30 \\\ \hline S_{n} & 2 & 8 & 18 & 32 & 50 & 72 & 98 & 128 \\\ \hline \end{array} $$ The completed table includes all the terms of the arithmetic sequence and their partial sums. The values of \(a_1 = 2\) and \(d = 4\) were used to compute these results.

Key concepts

Sequence of Partial Sums

When dealing with arithmetic series, one often encounters the term "sequence of partial sums." This simply refers to a sequence that represents the sum of the terms of an arithmetic sequence up to a given point. Let's break this down further.

If you have an arithmetic sequence, for example, 2, 6, 10, ..., then the sequence of partial sums would be the cumulative totals of these numbers as you progress. Here's how it goes:

• The first partial sum, denoted as \(S_1\), is simply the first term, so here, \(S_1 = 2\).
• The second partial sum, \(S_2\), is the sum of the first two terms: \(2 + 6 = 8\).
• The third partial sum, \(S_3\), adds the next term: \(8 + 10 = 18\).
• This process continues, creating a sequence of sums: 2, 8, 18, and so on.

Each total is built sequentially by adding the next term from the arithmetic sequence, resulting in a new partial sum. This helps in understanding how much the series accumulates over time.

Common Difference

The concept of a common difference is fundamental when dealing with arithmetic sequences and series. It is the constant value that separates consecutive terms in an arithmetic sequence. In simpler terms, it's the number you repeatedly add to get from one term to the next.

For the sequence in our exercise, 2, 6, 10, ..., the common difference \(d\) can be found by subtracting the first term from the second term. Here, \(d = 6 - 2 = 4\). This means that to find any term in the sequence, you can simply add 4 to the previous term.

• Starting from 2, adding 4 gives 6.
• Adding 4 to 6 gives 10.
• This process continues, maintaining a steady increase by 4 each time.

This constant addition is what makes the sequence "arithmetic" - it is equally spaced and predictable.
Understanding the common difference allows one to easily determine any term in an arithmetic sequence using the formula \(a_n = a_1 + (n-1)d\), which encodes the concept of common difference neatly.

Arithmetic Sequence

An arithmetic sequence is a list of numbers in which each term differs from the previous one by a fixed amount, known as the common difference. This structure makes arithmetic sequences predictable and easy to compute with, making them a cornerstone of mathematical study.

In the context of our exercise, the sequence: 2, 6, 10, ..., demonstrates the simple but powerful characteristics of an arithmetic sequence. Let's see how it works:

• Each term can be seen as a step up from the last, using the common difference, \(d = 4\).
• The general formula that describes any term \(a_n\) in an arithmetic sequence is \(a_n = a_1 + (n-1)d\), where \(a_1\) is the first term, and \(d\) is the common difference.
• So, to find any term in the sequence, you start with \(a_1\) and add \(d\), \((n-1)\) times.

This straightforward approach makes arithmetic sequences a fundamental topic in algebra and precalculus.
With these basic principles, any student can tackle problems involving arithmetic sequences and use them to find missing terms, analyze patterns, and solve real-life problems where uniform incrementation is involved.