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 sequence \(3,8,13,18, \ldots\) is $$ \begin{array}{c|c|c|c|c|c|c|c|c} \hline n & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline a_{n} & 3 & 8 & 13 & 18 & & & & \\ \hline S_{n} & 3 & 11 & 24 & 42 & & & & \\ \hline \end{array} $$

Short answer

Question: Find the value of \(a_6\) and \(S_7\) for the arithmetic sequence \(3, 8, 13, 18, \ldots\). Answer: \(a_6 = 28\) and \(S_7 = 126\).

Step-by-step solution

Identify the values of \(a_1\) and \(d\)

The first term of the sequence (\(a_1\)) is \(3\) and the difference between consecutive terms (\(d\)) is \(5\), since \(8-3=5\) and \(13-8=5\).

Fill in the missing terms (\(a_n\)) using the formula

Now, we will use the formula \(a_n = a_1 + (n-1)d\) to compute the missing terms of the sequence. For \(n = 6, 7,\) and \(8\), we have: - \(a_6 = a_1 + (6-1)d = 3 + 5(5) = 28\) - \(a_7 = a_1 + (7-1)d = 3 + 6(5) = 33\) - \(a_8 = a_1 + (8-1)d = 3 + 7(5) = 38\)

Calculate the partial sums (\(S_n\))

Lastly, we will use the formula \(S_n = \frac{n}{2}(a_1+a_n)\) to compute the partial sums. For \(n = 5, 6, 7,\) and \(8\), we have: - \(S_5 = \frac{5}{2}(3+a_5) = \frac{5}{2}(3+23) = \frac{5}{2}(26) = 65\) - \(S_6 = \frac{6}{2}(3+a_6) = \frac{6}{2}(3+28) = \frac{6}{2}(31) = 93\) - \(S_7 = \frac{7}{2}(3+a_7) = \frac{7}{2}(3+33) = \frac{7}{2}(36) = 126\) - \(S_8 = \frac{8}{2}(3+a_8) = \frac{8}{2}(3+38) = \frac{8}{2}(41) = 164\) Now, let's put everything together in the table: $$ \begin{array}{c|c|c|c|c|c|c|c|c} \hline n & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\\ \hline a_{n} & 3 & 8 & 13 & 18 & 23 & 28 & 33 & 38 \\\ \hline S_{n} & 3 & 11 & 24 & 42 & 65 & 93 & 126 & 164 \\\ \hline \end{array} $$

Key concepts

Arithmetic Sequence

An arithmetic sequence is a list of numbers where each number after the first is obtained by adding a fixed number, called the common difference, to the previous term. This fixed number is referred to as the common difference and is consistent throughout the sequence.
For example, in the sequence \(3, 8, 13, 18, \ldots\), the common difference \(d\) is 5, since each number is obtained by adding 5 to the previous number.
To find any term \(a_n\) in an arithmetic sequence, the formula used is:

• \(a_n = a_1 + (n - 1) \times d\)

Here, \(a_1\) is the first term and \(d\) is the common difference. By using this formula, you can compute any term in the sequence if you know the starting term and the common difference.

Partial Sums

Partial sums are a way of adding up the first few terms of a sequence to find their collective sum. In the context of an arithmetic sequence, partial sums are found by summing up the terms of the sequence up to a certain point.
For the sequence \(3, 8, 13, 18, \ldots\), the partial sums form another sequence \(S_1, S_2, S_3, \ldots\) where each term represents the sum of all terms up to that point, such as:

• \(S_1 = 3\)
• \(S_2 = 3 + 8 = 11\)
• \(S_3 = 3 + 8 + 13 = 24\)

The partial sum \(S_n\) of the first \(n\) terms can be calculated using the formula:

• \(S_n = \frac{n}{2} (a_1 + a_n)\)

This helps in calculating the total of any number of terms from the beginning of the sequence.

Additive Sequence

Additive sequences, like arithmetic sequences, involve a systematic addition of a constant number to get from one term to the next. However, generally speaking, not all additive sequences require the number to be constant.
In an arithmetic sequence, the fixed amount added between consecutive terms makes it a type of additive sequence. This makes understanding and calculation straightforward as each term is the previous one added by a constant \(d\).
This constant nature makes arithmetic sequences a fundamental concept in mathematics, as it forms a predictable pattern that can easily be worked with using basic arithmetic operations.

Series Formula

The series formula is an efficient way to sum up terms of an arithmetic sequence without having to manually add each term individually. This is particularly beneficial when dealing with long sequences.
For an arithmetic sequence, the series formula \(S_n = \frac{n}{2} (a_1 + a_n)\) helps find the sum of the first \(n\) terms. In this formula:

• \(n\) is the number of terms you want to add up
• \(a_1\) is the first term of the sequence
• \(a_n\) is the last term in the sequence up to which the sum is needed

This approach avoids the tedious process of adding each number individually, providing a quick and reliable method to obtain the sum of a series, harnessing the consistency of arithmetic sequences.