Problem 51 - Beginning and Intermediate Algebra

Question

Find \(S_{8}\) for each arithmetic sequence described below. $$a_{1}=-1, a_{8}=-29$$

Step-by-Step Solution

Answer

The sum of the first 8 terms of the arithmetic sequence, \(S_8\), is -120.

Step 1: 1. Find the common difference (d)

: We know that the arithmetic sequence formula is: \[a_n = a_1 + (n - 1)d\] We have \(a_1 = -1\), \(a_8 = -29\), and \(n = 8\). Plugging these values into the formula, we get: \[-29 = -1 + (8 - 1)d\] Now, solve for 'd'.

Step 2: 2. Solve for d

: Simplify the equation and isolate 'd': \[-29 = -1 + 7d\] \[-28 = 7d\] \[d = -4\] So, the common difference 'd' is -4.

Step 3: 3. Apply the sum formula

: Now that we know the first term, the last term, and the number of terms, we can find the sum of the first 8 terms using the formula for the sum of the first n terms of an arithmetic sequence: \[S_n = \frac{n}{2}(a_1 + a_n)\] Plugging the values into the formula, we have: \[S_8 = \frac{8}{2}(-1 - 29)\]

Step 4: 4. Calculate the sum

: Now, calculate the sum: \[S_8 = 4(-30)\] \[S_8 = -120\] So, the sum of the first 8 terms of the arithmetic sequence, \(S_8\), is -120.

Key Concepts

Arithmetic Series

An arithmetic series is a sum of terms that follow a specific pattern called an arithmetic sequence.
This means each term is created by adding a constant value to the previous one.
The pattern is easy to identify and let's us calculate the series' sum efficiently.

• The terms in an arithmetic series increase or decrease linearly (steadily).
• Every series has a 'first term' and a 'last term'.
• The way these terms progress is defined by the 'common difference'.

By knowing the sequence elements, we can use the formula for the *sum of an arithmetic series* to find the sum.
This is very handy, especially for sequences with many terms.

Common Difference

In arithmetic sequences, the 'common difference' is the key element that connects one term to the next.
It is what you add to a term to get to the next one.

• If the difference is positive, the sequence increases.
• If negative, the sequence decreases.
• If zero, all terms are the same.

To find the common difference, use the formula:\[ d = \frac{a_n - a_1}{n - 1} \].
In our example, calculating the common difference was crucial to understand the sequence behavior.
For the sequence starting with \(-1\) and ending with \(-29\), the common difference \(d\) was found to be \(-4\). This indicated that each term is \(4\) less than the previous one.

Sum of Terms

The sum of terms in an arithmetic sequence is called an arithmetic series.
To calculate the sum, use the specific formula for the arithmetic series.
For a sequence with a known number of terms \(n\), a first term \(a_1\), and a last term \(a_n\), the formula is:\[ S_n = \frac{n}{2} (a_1 + a_n) \].
This formula allows us to quickly find the total sum without adding each term manually.
In our exercise:

• The first term \(a_1\) was \(-1\).
• The last term \(a_8\) was \(-29\).
• We calculated the sum \(S_8\) for the first \(8\) terms to be \(-120\).

This approach makes it easy to handle larger sequences effectively.

Sequence Formula

The sequence formula is a tool that defines every term in an arithmetic sequence.
It expresses each term with respect to its position, first term, and common difference.

• The general formula is \(a_n = a_1 + (n - 1) d\).
• Where \(a_n\) is the nth term, \(a_1\) is the first term, and \(d\) is the common difference.

By using this, we can find any term in the sequence quickly.
For example, to find \(a_8\) given \(a_1 = -1\) and \(d = -4\), we used the formula to confirm it as \(-29\).
Understanding this formula helps predict terms and manage sequences systematically.