Problem 57 - Beginning and Intermediate Algebra

Question

Find \(S_{8}\) for each arithmetic sequence described below. $$a_{n}=-4 n-1$$

Step-by-Step Solution

Answer

The sum of the first 8 terms in the given arithmetic sequence is \(S_8 = -152\).

Step 1: Understand the arithmetic sequence formula and the given sequence

The formula for the nth term of an arithmetic sequence is: \(a_n = a_1 + (n-1)d\), where \(a_1\) is the first term, \(n\) is the position of the term in the sequence, and \(d\) is the common difference between consecutive terms in the sequence. For our given sequence, \(a_n = -4n - 1\), we can determine the common difference by finding the difference between consecutive terms.

Step 2: Find the common difference of the sequence

To find the common difference (\(d\)), substitute the values for consecutive terms in the given formula and find their difference: \(a_1 = -4(1) - 1 = -5\) \(a_2 = -4(2) - 1 = -9\) Now find the difference: \(d = -9 - (-5) = -4\) The common difference for this sequence is -4.

Step 3: Use the arithmetic series formula

The sum of an arithmetic series can be found using the formula: \[S_n = \frac{n(a_1 + a_n)}{2}\] Here, we need to find \(S_8\), so we need to determine the value of \(a_8\): \(a_8 = -4(8) - 1 = -33\) Now we can find the sum of the first 8 terms:

Step 4: Calculate the sum of the first 8 terms

Plug the values into the arithmetic series formula: \(S_8 = \frac{8(-5 +(-33))}{2}\) Simplify the expression: \(S_8 = \frac{8(-38)}{2} = \frac{-304}{2} = -152\) So the sum of the first 8 terms in the given arithmetic sequence is -152.

Key Concepts

Common Difference

In an arithmetic sequence, the common difference is a very important component. It is the consistent amount added to each term to reach the next term in the sequence. Identifying this helps in understanding the sequence's pattern.
To determine the common difference ((d)), we calculate the difference between two consecutive terms. In the given sequence, which is defined by \(a_n = -4n - 1\), we find:

• The first term, \(a_1 = -4(1) - 1 = -5\)
• The second term, \(a_2 = -4(2) - 1 = -9\)

By subtracting these, \(d = a_2 - a_1 = -9 - (-5) = -4\).
This consistent shift of -4 indicates each term decreases by 4 as we move along the sequence. Recognizing the common difference helps simplify the calculation of any term within the sequence.

N-th Term Formula

The n-th term formula of an arithmetic sequence allows us to calculate any term's value based on its position \(n\) in the sequence. The general formula is:\[a_n = a_1 + (n-1)d\]Here,

• \(a_1\) is the first term
• \(d\) is the common difference
• \(n\) denotes the specific term number

For our sequence, it's easier as we have the direct expression \(a_n = -4n - 1\), which indicates each position n multiplies with -4 and subtracts 1.
This formula is useful to calculate the desired term without sequentially adding the common difference. For example, to find \(a_8\):\[-4(8) - 1 = -33\]With this n-th term formula, we efficiently target the required term directly from its position number.

Arithmetic Series Formula

When we talk about the arithmetic series formula, we are interested in calculating the sum of a specified number of terms in an arithmetic sequence. The formula for the sum \(S_n\) of the first \(n\) terms is:\[S_n = \frac{n(a_1 + a_n)}{2}\]Here,

• \(n\) is the number of terms
• \(a_1\) is the first term of the sequence
• \(a_n\) is the n-th term

In this specific problem, we're finding \(S_8\), the sum of the first eight terms. We already know:

• \(a_1 = -5\)
• \(a_8 = -33\)
• \(n = 8\)

So plugging into the formula:\[S_8 = \frac{8(-5 + (-33))}{2} = \frac{8(-38)}{2}\]This gives us a streamlined way to accumulate the values of multiple terms.

Sum of Terms

Calculating the sum of terms in an arithmetic sequence can provide insights into the overall pattern and value created by the sequence. Using the arithmetic series formula, we determine this cumulative amount.
The formula \(S_n = \frac{n(a_1 + a_n)}{2}\) is employed to find out the sum of the sequence up until a certain point, \(n\). By accurately identifying the terms \(a_1\) and \(a_n\), and the number of terms \(n\), you can easily compute the sum without listing all terms individually.
In the specific problem at hand, we found:

• \(n = 8\), signifying the first 8 terms
• \(a_1 = -5\)
• \(a_8 = -33\)

This gives us the sum as:\[S_8 = \frac{8(-38)}{2} = -152\]Trusting your comprehension of sequences and differences, these concepts seamlessly reveal substantial numeric results without unnecessary complexity.