Problem 53 - Beginning and Intermediate Algebra

Question

Find \(S_{8}\) for each arithmetic sequence described below. $$a_{1}=3, d=5$$

Step-by-Step Solution

Answer

The sum of the first 8 terms of the given arithmetic sequence, \(S_8\), is 164.

Step 1: Find the 8th term (\(a_8\)) of the arithmetic sequence

To find the 8th term of the arithmetic sequence, we can use the formula: \[a_n = a_1 + (n-1)d\]. In this exercise, we are given the first term \(a_1 = 3\), common difference \(d = 5\), and the term we want to find is the 8th term, so \(n = 8\). So, we have: \[a_8 = a_1 + (n-1)d = 3 + (8-1)(5) = 3 + 7(5)\]

Step 2: Complete the calculation for \(a_8\)

Now we calculate the value of \(a_8\): \[a_8 = 3 + 7(5) = 3 + 35 = 38 \]

Step 3: Apply the arithmetic series formula

Now we can find the sum of the first 8 terms of the arithmetic sequence using the formula: \[S_n = \frac{n(a_1 + a_n)}{2}\], where \(n=8\), \(a_1 = 3\) and \(a_8 = 38\). So, we have: \[S_8 = \frac{8(3 + 38)}{2}\]

Step 4: Calculate the result

Now find the value of \(S_8\): \[S_8 = \frac{8(3 + 38)}{2} = \frac{8(41)}{2} = \frac{328}{2} = 164 \] The sum of the first 8 terms of the given arithmetic sequence is 164.

Key Concepts

arithmetic series

An arithmetic series is essentially the sum of the terms in an arithmetic sequence. Imagine taking a sequence where each term increases or decreases by a constant amount and adding all those terms together. This series gives us the total sum from the first term to any desired term of the sequence.

To compute an arithmetic series, there's a handy formula:

• \[ S_n = \frac{n(a_1 + a_n)}{2} \]

Here, \( S_n \) represents the sum of the first \( n \) terms, \( a_1 \) is the first term, and \( a_n \) is the \( n \)-th term. This formula simplifies calculations tremendously by averaging the sum of the first and last terms and multiplying by the number of terms.

common difference

The common difference is a crucial part of an arithmetic sequence. It is the consistent interval between consecutive terms. Think of it as the "step" you take to move from one term to the next within the sequence.

The common difference can be derived by subtracting any term from the following term. Mathematically, if you have two consecutive terms \( a_{n+1} \) and \( a_n \), the common difference \( d \) is

• \[ d = a_{n+1} - a_n \]

Understanding the common difference lets us predict future terms and find the total number of terms needed when solving for items like the sum of an arithmetic series.

first term

The first term of an arithmetic sequence, often denoted as \( a_1 \), is the starting point of the sequence. It sets the sequence rolling.

In any arithmetic equation or series, knowing the first term is essential because it serves as the baseline from which all other terms in the sequence are calculated. This first number anchors the sequence and helps in using formulas to find specific terms or sums of terms efficiently. For instance, when finding the sum of terms in an arithmetic series, \( a_1 \) is pivotal in using the series formula correctly.

sum of terms

When we talk about the sum of terms in an arithmetic series, we're referring to adding together several specific terms from the sequence. Calculating the sum of terms can be easily done using the arithmetic series formula.

The formula

• \[ S_n = \frac{n(a_1 + a_n)}{2} \]

is efficient because it allows calculating the sum without adding each term individually. You utilize the first and the last terms in the desired range of the sequence to compute the total. This approach saves time and reduces error, particularly when dealing with long sequences. In the provided exercise, this formula was used to find that the sum of the first 8 terms of the sequence was 164, emphasizing the power of using a well-structured method.