Problem 59 - Beginning and Intermediate Algebra

Question

Find \(S_{8}\) for each arithmetic sequence described below. $$a_{n}=3 n+4$$

Step-by-Step Solution

Answer

The sum of the first 8 terms of the arithmetic sequence is \(S_8 = 140\).

Step 1: Understand the arithmetic sequence formula

In this exercise, we are given the arithmetic sequence formula as \(a_n = 3n+4\). Using this formula, we can find the value of any term in the sequence by plugging in the value of \(n\).

Step 2: Calculate the first 8 terms

Now we will find the value of each term up to the 8th term, by plugging in \(n\) from 1 to 8 in the formula \(a_n = 3n + 4\). (1) \(a_1 = 3(1) + 4 = 7\) (2) \(a_2 = 3(2) + 4 = 10\) (3) \(a_3 = 3(3) + 4 = 13\) (4) \(a_4 = 3(4) + 4 = 16\) (5) \(a_5 = 3(5) + 4 = 19\) (6) \(a_6 = 3(6) + 4 = 22\) (7) \(a_7 = 3(7) + 4 = 25\) (8) \(a_8 = 3(8) + 4 = 28\)

Step 3: Find the sum of the first 8 terms

Now, we will add all the terms we found in Step 2 to find the sum \(S_8\). $$S_8 = a_1 + a_2 + a_3 + a_4 + a_5 + a_6 + a_7 + a_8 = 7 + 10 + 13 + 16 + 19 + 22 + 25 + 28 = 140$$ So, the sum of the first 8 terms of the arithmetic sequence is \(S_8 = 140\).

Key Concepts

Sequence Formula

An arithmetic sequence is a series of numbers in which the difference between consecutive terms remains constant. This difference is called the "common difference." To define or express each term in such a sequence, we use the sequence formula. In our problem, this formula is given as \(a_n = 3n + 4\). Here, \(n\) represents the term number you're interested in.

• The sequence formula: It's a linear equation where each term can be calculated individually. It's set in the form of \(a_n = a_1 + (n-1) \, d\), where \(a_1\) is the first term and \(d\) is the common difference.
• Common difference calculation: In \(a_n = 3n + 4\), the common difference is 3, which we derive from the coefficient of \(n\).

Notice how using the formula simplifies finding any term, such as the 100th term, without having to calculate all previous terms.

Sum of Terms

The sum of terms in an arithmetic sequence can be calculated in a systematic way using a formula. This is crucial when dealing with a long list of numbers.

• Simple addition: To find the sum, like in our problem with the first 8 terms, you add up each calculated term.
• Using the formula: Alternatively, the sum formula for an arithmetic sequence is \(S_n = \frac{n}{2} (a_1 + a_n)\), where \(S_n\) is the sum of the first \(n\) terms, \(a_1\) is the first term, and \(a_n\) is the nth term.

With this approach, you can swiftly find the total sum without computing each term when dealing with large sequences.

Series Calculation

When we talk about the series in arithmetic sequences, we're mostly concerned with finding the total of a certain number of terms. This process is known as series calculation.

• Series definition: It's the sum of the terms in a sequence up to a specific point.
• The arithmetic series: Often expressed as the sum of the sequence, employs formulas to ease calculation. For example, handling an arithmetic sequence of 100 terms becomes simple using the series sum formula \(S_n\).

This calculation is essential in various real-life scenarios where predicting long-term outcomes or averages is required.

Linear Expression

A linear expression is a mathematical statement where each term is either a constant or the product of a constant and a single variable. In the context of arithmetic sequences, our sequence formula \(a_n = 3n+4\) is itself a linear expression.

• Understanding linear expressions: It's characterized by variables raised only to the first power. The graph of a linear expression is always a straight line.
• Role in sequences: It shows a constant rate of change between terms, which defines the essence of an arithmetic sequence.

Recognizing and working with linear expressions in sequences aids in solving a wide range of mathematical problems, offering a foundational structure for understanding how sequences develop.