Problem 28 - Algebra 2

Question

Find \(a_{1}\) for each arithmetic series described. $$ d=0.5, n=31, S_{31}=573.5 $$

Step-by-Step Solution

Answer

The first term, \( a_1 \), is 11.

Step 1: Understanding the Arithmetic Series Formula

The sum of an arithmetic series can be found using the formula: \( S_n = \frac{n}{2} (a_1 + a_n) \), where \( S_n \) is the sum of the first \( n \) terms, \( a_1 \) is the first term, and \( a_n \) is the nth term.

Step 2: Finding Expression for the Nth Term

The nth term, \( a_n \), in an arithmetic sequence can be found using: \( a_n = a_1 + (n-1) \cdot d \). Here, \( d \) is the common difference, which is given as \(0.5\). Thus, \( a_{31} = a_1 + 30 \times 0.5 \).

Step 3: Setting Up the Equation with Known Values

Substitute \( n = 31 \), \( S_{31} = 573.5 \), and the expression for \( a_{31} \) into the sum formula: \( 573.5 = \frac{31}{2} (a_1 + a_1 + 15) \).

Step 4: Simplifying and Solving the Equation

Simplify the equation: \( 573.5 = 31(a_1 + 7.5) \). Divide both sides by 31: \( a_1 + 7.5 = 18.5 \). Subtract 7.5 from both sides to solve for \( a_1 \): \( a_1 = 18.5 - 7.5 \).

Step 5: Final Calculation

Perform the final calculation: \( a_1 = 11 \).

Key Concepts

Sum of Series Formula

The sum of an arithmetic series is an important concept that helps us find the total of several numbers arranged in a sequence. You can determine the sum using a simple formula: - \( S_n = \frac{n}{2} (a_1 + a_n) \). Here, \( S_n \) represents the sum of the first \( n \) terms in the series. - \( a_1 \) is the first term, and - \( a_n \) is the \( n \)th term of the sequence.
This formula is particularly useful because it depends only on the first and last terms of the sequence, making calculations faster and easier. By applying this formula, we can quickly add up all the terms without calculating each one individually.
This is exactly what we did in this exercise when we were asked to find the first term \( a_1 \). By knowing the sum \( S_{31} \), the number of terms \( n \), and utilizing our formula, we set the stage to solve for other unknowns.

Common Difference

In arithmetic sequences, the common difference \( d \) is the key to finding the relationship between consecutive terms. Every term is created by adding the common difference to the previous term, which helps in maintaining a uniform pattern. The common difference can be defined as follows: - For two consecutive terms \( a_n \) and \( a_{n+1} \), the common difference is \( d = a_{n+1} - a_n \).
In the exercise, we are given \( d = 0.5 \), signaling the constant increment from one term to the next. Understanding this concept allowed us to develop the expression for the nth term and helped us solve for the first term \( a_1 \). This kind of problem is straightforward once you understand the consistent steps involved in building the sequence.

First Term Calculation

Finding the first term \( a_1 \) is often necessary to understand the whole sequence when some information is already given, like the sum of the series or the common difference. In the exercise, we started by using the sum formula along with the expression for the nth term.Here's how it unfolds: 1. We first formulated the expression for \( a_{31} \) using \( a_n = a_1 + (n-1) \cdot d \).2. Solving means, plug in \( n=31 \) and \( d=0.5 \), leading to \( a_{31} = a_1 + 15 \).3. Then, substitute \( a_{31} \) into the sum formula to establish an equation \( 573.5 = \frac{31}{2}(a_1 + a_1 + 15) \).4. Simplifying this lets us solve the equation to find \( a_1 \).5. Finally, we calculated \( a_1 = 11 \), allowing us to define the entire sequence accurately.

Nth Term Expression

The nth term expression shows any term's value in an arithmetic sequence, making it crucial for finding specific points within the series. It is written as:- \( a_n = a_1 + (n-1) \cdot d \)This formula allows us to calculate any term if we know the first term \( a_1 \) and the common difference \( d \).
In the exercise, we used this expression to find \( a_{31} \). By substituting known values, we derived: - \( a_{31} = a_1 + 30 \times 0.5 \).- After solving this alongside the sum of the series formula, we determined the exact value for \( a_1 \).
Understanding and applying this formula is straightforward and vital for solving such arithmetic problems. It connects directly to the sum formula, proving essential for extracting relevant information about the series.