Problem 60 - Beginning and Intermediate Algebra

Question

Find \(S_{8}\) for each arithmetic sequence described below. $$a_{n}=-6 n+5$$

Step-by-Step Solution

Answer

The sum of the first 8 terms of the arithmetic sequence is \(S_8 = -176\).

Step 1: Find the first 8 terms of the sequence

To find the first 8 terms of the sequence, we will substitute the values of n from 1 to 8 into the formula: \(a_1 = -6(1) + 5\) \(a_2 = -6(2) + 5\) \(a_3 = -6(3) + 5\) \(a_4 = -6(4) + 5\) \(a_5 = -6(5) + 5\) \(a_6 = -6(6) + 5\) \(a_7 = -6(7) + 5\) \(a_8 = -6(8) + 5\) After calculating the values of the terms, we get: \(a_1 = -1\) \(a_2 = -7\) \(a_3 = -13\) \(a_4 = -19\) \(a_5 = -25\) \(a_6 = -31\) \(a_7 = -37\) \(a_8 = -43\)

Step 2: Calculate S_8, the sum of first 8 terms

To calculate the sum of the first 8 terms, we can use the formula: \(S_n = \frac{n (a_1 + a_n)}{2}\) For our problem, we want to find \(S_8\), so we will substitute \(n = 8\) and the values of \(a_1\) and \(a_8\): \(S_8 = \frac{8((-1) + (-43))}{2}\) \(S_8 = \frac{8(-44)}{2}\) \(S_8 = \frac{-352}{2}\) \(S_8 = -176\) Hence, the sum of the first 8 terms of the arithmetic sequence is \(-176\).

Key Concepts

Sum of Sequence

When working with arithmetic sequences, calculating the sum of a series of terms is a fundamental step. The formula for finding the sum of the first \(n\) terms, which is the sequence's sum (denoted as \(S_n\)), is structured to be both simple and efficient. It relies on the number of terms \(n\), the first term \(a_1\), and the last term \(a_n\) in the series:

• The formula is: \(S_n = \frac{n (a_1 + a_n)}{2}\).

This formula operates by calculating the mean of the first and the last term, then multiplying by the number of terms. This method efficiently considers the symmetry and constant difference in an arithmetic sequence, bypassing the need to add each term manually.

As shown in the solution, for our sequence, by substituting \(a_1 = -1\) and \(a_8 = -43\), along with \(n = 8\), the calculated sum \(S_8\) was determined to be \(-176\). This negative value aligns with the sequence's pattern of decreasing numbers.

Formula for nth Term

The formula for determining the \(n\)th term in an arithmetic sequence is crucial because it encapsulates the sequence's consistent pattern. Arithmetic sequences have a common difference between consecutive terms, and the general formula is expressed as:

• \(a_n = a_1 + (n-1)d\)

Where:

• \(a_n\) is the \(n\)th term,
• \(a_1\) is the first term of the sequence,
• \(d\) is the common difference between consecutive terms.

However, for this specific sequence, the formula given was \(-6n + 5\), which directly calculates the \(n\)th term without first calculating the common difference. This linear expression lets us input any term number \(n\) to get the corresponding sequence term instantly, as demonstrated for terms \(a_1\) through \(a_8\). Each calculated term fits within the overall sequence behavior of decreasing by the common difference.

Sequence Terms Calculation

Calculating individual sequence terms involves substituting \(n\) into the formula provided. For the problem solved, this was repeated for each integer from 1 to 8 to find the respective terms of the sequence.

The process involves understanding:

• Start from \(n=1\) and move sequentially upwards.
• For each \(n\), replace it in the formula \(-6n + 5\).
• Solve the resulting equation to find the sequence term.

Implementing this process with \(n=1\) gives \(a_1 = -6(1) + 5 = -1\). Continuing the same for \(n=2\) through \(n=8\) yields the terms \(-7, -13, -19, -25, -31, -37, -43\) respectively. Recognizing this pattern is key to comprehending how arithmetic sequences function, allowing one to accurately predict all terms, given the sequence rule or expression.