Problem 55 - Algebra 2

Question

Write an expression for the sum of a 6 -term arithmetic sequence with first term of 3 and a common difference of 4 . Then find the sum.

Step-by-Step Solution

Answer

The sum of the arithmetic sequence is \(S_6 = 78\).

Step 1: Substitute the values into the sum formula

Substitute \(a = 3, d=4\), and \(n=6\) into the formula for the sum of an arithmetic series \(S_n = n/2(2a + (n-1)d)\). Thus, the expression becomes \(S_6 = 6/2(2*3 + (6-1)*4)\).

Step 2: Simplify the Equation

Multiply out the expression to make it easier to calculate. Lt leads to \(S_6 = 3*(6 + 20)\).

Step 3: Calculate the sum

Now calculate the sum by further simplifying the expression \(S_6 = 3*26\).

Key Concepts

Sum of Arithmetic Series

An arithmetic series is simply the sum of the terms in an arithmetic sequence. Calculating this sum is a straightforward process when you know the formula. The formula for finding the sum of the first \(n\) terms in an arithmetic sequence is given by: \[ S_n = \frac{n}{2} (2a + (n-1) d) \]where:

• \(S_n\) is the sum of the series,
• \(n\) is the number of terms you want to sum up,
• \(a\) is the first term of the sequence,
• \(d\) is the common difference.

This formula helps by converting the repetitive addition of numbers into a neat calculation. For instance, if you want to find the sum of a 6-term series with a first term of 3 and a common difference of 4, you plug these values into the formula to find \(S_6\).

Common Difference

In an arithmetic sequence, the common difference is a crucial constant that defines the consistent interval between consecutive terms. This difference is always the same, and you can find it by subtracting any term from the term that comes right after it. Mathematically, if the sequence is \(a_1, a_2, a_3, \ldots\), the common difference \(d\) can be expressed as:\[ d = a_2 - a_1 \]This difference ensures that the sequence maintains a uniform progression. In the exercise given, the common difference is given as 4, which means every term increases by 4 units from the previous term. Understanding the common difference is essential for predicting future terms or finding the sum of a series.

First Term

The first term in an arithmetic sequence, often denoted as \(a\), is pivotal as it sets the starting point for the entire sequence. From this point, each subsequent term is derived by adding the common difference \(d\). For example, if your first term is 3, like in the given exercise, your sequence begins here, and each following term increases by the common difference.

• If \(a = 3\) and \(d = 4\), the sequence would be: 3, 7, 11, 15, 19, 23.

Recognizing the first term's role helps in organizing and predicting the flow of the sequence correctly. It's particularly important when you apply the sum formula for the series because it's part of the calculation to determine the overall sum.

Sequence Formula

The sequence formula for an arithmetic sequence is integral, as it helps to find any term in the sequence without listing all the previous terms. This formula is given by:\[ a_n = a + (n-1) d \]where:

• \(a_n\) is the nth term you wish to find,
• \(a\) is the first term,
• \(n\) is the term number,
• \(d\) is the common difference.

Using this formula, you can quickly determine any term's position in the sequence. It's particularly useful when dealing with large sequences where listing each term becomes tedious. For example, to find the 6th term in the sequence starting with 3 and a common difference of 4, you substitute into the formula to find: \( a_6 = 3 + (6-1) \times 4 = 23 \). This saves time and ensures accuracy.