Problem 68 - Algebra 2

Question

Find the sum of each arithmetic series. $$ 50+44+38+\dots+8 $$

Step-by-Step Solution

Answer

The sum of the series is 232.

Step 1: Identify the Arithmetic Sequence

First, observe the given arithmetic series: 50, 44, 38, ..., 8. Notice that we need to find the common difference and the number of terms in the sequence. The first term, \( a \), is 50.

Step 2: Determine the Common Difference

To find the common difference \( d \), subtract the second term from the first term: \( d = 44 - 50 = -6 \).

Step 3: Find the Number of Terms

Use the formula for the last term of an arithmetic sequence \( a_n = a + (n-1)d \). Here, \( a_n = 8 \). Substitute the values: \[ 8 = 50 + (n-1)(-6) \]. Simplify the equation to find \( n \): \[ 8 = 50 - 6n + 6 \]. \[ 8 = 56 - 6n \]. \[ 6n = 56 - 8 \]. \[ 6n = 48 \]. \[ n = 8 \]. So, there are 8 terms.

Step 4: Apply the Sum Formula for Arithmetic Series

The sum \( S_n \) of an arithmetic series can be found using the formula \( S_n = \frac{n}{2}(a + a_n) \). Substitute the values: \[ S_8 = \frac{8}{2}(50 + 8) \]. \[ S_8 = 4 \times 58 \].

Step 5: Calculate the Sum

Perform the calculation: \[ S_8 = 232 \]. This is the sum of the series.

Key Concepts

Common Difference

In an arithmetic sequence, the **common difference** is a key element that defines the entire sequence. It's the constant amount by which each term in the sequence increases or decreases from the previous one. To find it, you simply subtract the first term from the second term.

• In our example, the sequence starts with 50, followed by 44, and then 38.
• Calculating the common difference: 44 - 50 gives us -6.

This value of -6 tells us that each subsequent term in the sequence decreases by 6. Understanding this difference helps to predict and calculate any term within the series.

Number of Terms

Calculating the **number of terms** in an arithmetic sequence is essential to evaluating its sum. This count tells us how many terms are making up the sequence or series.
For our series, we use the last term of the sequence equation:\[ a_n = a + (n-1)d \]Where:

• \( a \) is the first term of the sequence
• \( d \) is the common difference
• \( a_n \) is the last term provided

By using 50 as the first term and 8 as the last term with a common difference of -6, we solve for the total number of terms \( n \). After substituting these values and solving, we find that our sequence contains 8 terms.

Sum of Arithmetic Series

The **sum of an arithmetic series** can be calculated using a specific formula that simplifies the addition of all terms in the sequence. The formula for the sum \( S_n \) is:\[ S_n = \frac{n}{2}(a + a_n) \]Here, \( n \) is the number of terms, \( a \) is the first term, and \( a_n \) is the last term.

• For our series, the first term \( a \) is 50, the last term \( a_n \) is 8, and \( n \) is 8.

By plugging these figures into the formula, we calculate the sum: \( S_8 = \frac{8}{2}(50 + 8) \), which equals 232. This makes finding the sum of a potentially long list much faster and easier.

Arithmetic Sequence

An **arithmetic sequence** is a sequence of numbers in which the difference between consecutive terms is constant. This regular pattern makes it possible to predict subsequent numbers and efficiently perform calculations involving the sequence.Characteristics of an arithmetic sequence include:

• Identifiable first term, often denoted as \( a \)
• A constant common difference, \( d \), between successive terms
• The ability to describe any term using the formula: \( a_n = a + (n-1)d \)

The sequence from our problem, with terms starting at 50 and decreasing by 6 each time, demonstrates these principles perfectly. Understanding these concepts helps in effectively managing sequences and series.