Problem 23 - College Algebra with Modeling and Visualization

Question

The sum of an arithmetic series with 15 terms is \(255 .\) If \(a_{1}=3,\) find \(a_{15}.\)

Step-by-Step Solution

Answer

The 15th term \( a_{15} \) is 31.

Step 1: Understand the Arithmetic Series Formula

The formula for the sum of an arithmetic series is given by \( S_n = \frac{n}{2} \times (a_1 + a_n) \), where \( S_n \) is the sum of the series, \( n \) is the number of terms, \( a_1 \) is the first term, and \( a_n \) is the last term.

Step 2: Identify the Given Values

From the exercise, we know \( n = 15 \), \( a_1 = 3 \), and \( S_{15} = 255 \). We need to find \( a_{15} \), which is the last term \( a_n \).

Step 3: Substitute the Values into the Formula

Substitute the known values into the sum formula: \( 255 = \frac{15}{2} \times (3 + a_{15}) \).

Step 4: Simplify the Equation

Simplify the equation: \( 255 = \frac{15}{2} \times (3 + a_{15}) \), multiply both sides by 2 to eliminate the fraction: \( 510 = 15 \times (3 + a_{15}) \).

Step 5: Solve for \( a_{15} \)

Divide both sides by 15 to isolate the expression with \( a_{15} \): \( 34 = 3 + a_{15} \). Subtract 3 from both sides: \( a_{15} = 31 \).

Key Concepts

Sum of Arithmetic Series

Understanding the sum of an arithmetic series is essential when working on various mathematical problems. An arithmetic series is essentially just the addition of a sequence of numbers where the difference between consecutive terms is constant. This difference is known as the common difference.
To find the sum of an arithmetic series, we use the formula:

• \[ S_n = \frac{n}{2} \times (a_1 + a_n) \]
• Where \(S_n\) represents the sum of the series, \(n\) is the total number of terms, \(a_1\) is the first term, and \(a_n\) is the last term.

This formula helps us efficiently calculate the sum without needing to add each term individually, especially helpful in long sequences. In the given exercise, the sum of 15 terms was calculated using this formula by integrating the given values directly into the expression. This approach demonstrates how an arithmetic series can be manipulated to find unknown terms while simplifying calculations.

Number of Terms in Series

The number of terms in an arithmetic series is a crucial element that directly influences the calculation of its sum. Often denoted by \(n\), this number tells us how many terms are being added together in the series.
In the earlier example, \(n = 15\) indicated that there were 15 numbers in total included in the series sum. This number not only affects the arithmetic series formula but also influences how we understand the series structure as a whole.
Calculating the number of terms correctly is crucial because it has a proportionate impact when we apply it in the sum formula as seen:

• Each term contributes to the overall sum through the multiplication by half the number of terms, thus emphasizing the arithmetic mean of the first and last terms.

Understanding the number of terms aligns with recognizing the structure and finding other unknowns within the sequence, like the final term in a given series. This understanding allows for flexibility in deriving necessary values using different approaches.

Arithmetic Series Formula

The arithmetic series formula is the cornerstone for solving problems related to arithmetic sequences. Using this formula, we can solve for various unknowns such as the sum of the series or the terms involved.
Mathematically, the formula is expressed as:

• \[ S_n = \frac{n}{2} \times (a_1 + a_n) \]
• Where \(S_n\) is the total sum of the series, \(n\) represents the number of terms, \(a_1\) is the initial term, and \(a_n\) is the final term in the sequence.

This formula proves incredibly useful in class exercises where certain elements like the last term or the sum are unknown. By rearranging and substituting known values, all other parts of the formula can be solved.
In the exercise given, the formula allowed for the determination of \(a_{15}\) which was unknown initially. After substituting the known values \(n = 15\), \(a_1 = 3\), and \(S_n = 255\) into the formula, deductive algebraic manipulation helped isolate and resolve the unknown term. This practical use of the arithmetic series formula illustrates its versatility and significance in arithmetic calculations.