Problem 44 - Algebra 2

Question

Find \(S_{n}\) for each arithmetic series described. $$ a_{1}=43, n=19, a_{n}=115 $$

Step-by-Step Solution

Answer

The sum of the series is 1501.

Step 1: Identify Formula for Sum of Arithmetic Series

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

Step 2: Substitute Given Values into Formula

Using the formula \( S_n = \frac{n}{2} (a_1 + a_n) \) and the values \(a_1 = 43\), \(n = 19\), and \(a_n = 115\), substitute them into the formula: \[ S_{19} = \frac{19}{2} (43 + 115) \]

Step 3: Simplify Inside the Parentheses

Calculate the sum inside the parentheses: \(43 + 115 = 158\). The expression now becomes: \[ S_{19} = \frac{19}{2} \, (158) \]

Step 4: Calculate the Product and Division

Compute \(158\) multiplied by \(19\), and then divide by \(2\): \[ S_{19} = \frac{19 \, \times \, 158}{2} \] Calculate \(19 \, \times \, 158 = 3002\) and then \(\frac{3002}{2} = 1501\).

Step 5: Final Result

The sum of the first \(19\) terms of the arithmetic series is \(S_{19} = 1501\).

Key Concepts

Sum of Arithmetic Series

Understanding the concept of an arithmetic series is important, as it allows us to compute the sum of a sequence of numbers that increase by a constant difference. The formula used to find the sum of an arithmetic series is:\[S_n = \frac{n}{2} (a_1 + a_n)\]Here, \(S_n\) represents the sum of the first \(n\) terms of the series. This formula efficiently combines the first term \(a_1\) and the \(n\)-th term \(a_n\), allowing us to calculate the sum without having to add each individual term manually.
This is particularly useful when dealing with a large number of terms, keeping calculations simple and manageable. When substituting the specific values of the first term, \(n\), and the last term into this formula, you can quickly discover the total sum of the series. This method is both concise and easy to apply, especially with practice.

Arithmetic Sequence Formula

An arithmetic sequence is a list of numbers with a common difference between consecutive terms. To identify each term in an arithmetic sequence, you use the formula:
\[a_n = a_1 + (n-1)d\]

• \(a_n\) - the \(n\)-th term
• \(a_1\) - the first term
• \(n\) - the term number in the sequence
• \(d\) - the common difference between terms

The significance of this formula lies in its ability to pinpoint any term in the sequence without having to successively add the common difference.
This results in a direct calculation of any term's value based on its position \(n\), simplifying the process of finding terms within a sequence. Combining this understanding with the sum formula provides strong tools to tackle problems related to arithmetic sequences and series.

Series Calculation Steps

Computing the sum of an arithmetic series involves a series of orderly steps:
**Identify the Formula:** Knowing the correct formula is the first step. For an arithmetic series, this is \(S_n = \frac{n}{2} (a_1 + a_n)\).
**Inserting Known Values:** With the given values—\(a_1 = 43\), \(n = 19\), and \(a_n = 115\)—substitute into the formula. This gives \(S_{19} = \frac{19}{2} (43 + 115)\).
**Parentheses Simplification:** Calculate the expression inside the parentheses: \(43 + 115 = 158\). This changes the equation to \(S_{19} = \frac{19}{2} \, (158)\).
**Final Calculation:** Multiply and divide as required: \(19 \, \times \, 158 = 3002\), followed by \(\frac{3002}{2} = 1501\).
By meticulously following each step, the solution becomes clear and systematic. Ensuring accuracy at each phase leads to a successful computation of the series sum, which in this case is 1501.