Problem 27 - Algebra 2

Question

Find \(a_{1}\) for each arithmetic series described. $$ d=-4, n=42, S_{42}=-3360 $$

Step-by-Step Solution

Answer

The first term \(a_1\) is 2.

Step 1: Understand the Given Information

We are given an arithmetic series with the common difference \(d = -4\), the number of terms \(n = 42\), and the sum of the series \(S_{42} = -3360\). We need to find the first term of the series \(a_1\).

Step 2: Use the Sum Formula for an Arithmetic Series

Recall the formula for the sum of an arithmetic series: \[ S_n = \frac{n}{2} (a_1 + a_n) \]Substituting the known values, the equation becomes:\[ -3360 = \frac{42}{2} (a_1 + a_{42}) \]

Step 3: Simplify and Solve for \(a_1 + a_{42}\)

Calculate \(\frac{42}{2} = 21\) and rewrite the equation:\[ -3360 = 21(a_1 + a_{42}) \]Divide both sides by 21 to isolate \(a_1 + a_{42}\):\[ a_1 + a_{42} = \frac{-3360}{21} = -160 \]

Step 4: Use the nth-Term Formula to Find \(a_{42}\)

The nth-term of an arithmetic series is given by:\[ a_n = a_1 + (n-1)d \]Substituting \(d = -4\), \(n = 42\), and \(n-1 = 41\), we get:\[ a_{42} = a_1 + 41(-4) \]Simplify to find \(a_{42}\):\[ a_{42} = a_1 - 164 \]

Step 5: Solve for \(a_1\)

Substitute \(a_{42} = a_1 - 164\) into the equation from Step 3:\[ a_1 + (a_1 - 164) = -160 \]Simplify and solve for \(a_1\):\[ 2a_1 - 164 = -160 \]Add 164 to both sides:\[ 2a_1 = 4 \]Divide by 2 to find \(a_1\):\[ a_1 = 2 \]

Key Concepts

Common Difference

An arithmetic series is a sequence of numbers where each term after the first is generated by adding a constant number, known as the common difference, to the previous term. The common difference, denoted as \(d\), helps establish the relationship between terms in the sequence. For example, if the common difference is \(-4\), as in the given problem, each term is \(-4\) less than the term before it. This pattern continues throughout the sequence.
Understanding the common difference is crucial when dealing with arithmetic progressions, as it allows us to find any term or the total sum over a set number of terms. It simplifies predicting future terms without having to add each step manually. When the difference is negative, the series decreases with each term, forming a descending pattern.

Sum Formula

The sum of an arithmetic series refers to the total of all the terms added together, often represented by \(S_n\) for \(n\) terms. The formula to calculate this is:

• \(S_n = \frac{n}{2} (a_1 + a_n)\)

This formula accounts for the average of the first and last term, multiplied by the number of terms to give the sum. In the problem at hand, we have \(S_{42} = -3360\), with 42 terms in total, meaning this series is providing a negative sum.
The provided sum allows students to work backward using the known values, such as the total number of terms and the sum, to find unknowns like the first term \(a_1\). Understanding the sum formula enables extrapolation and verification of series relations by using given data efficiently.

Nth-term Formula

An essential tool in exploring arithmetic sequences is the nth-term formula. It allows us to find any term in the sequence without listing all preceding numbers. The formula is:

• \(a_n = a_1 + (n-1)d\)

In this formula, \(a_1\) is the first term, and \(d\) is the common difference. It provides the necessary progression from the first term to the particular term we want. For example, to find \(a_{42}\), the 42nd term:

• \(a_{42} = a_1 + 41(-4)\)

Here, \(41\) is derived from \(n-1\), indicating the number of steps from the first to the desired term. Differentiating how terms are spaced lets students solve for missing elements in the sequence quickly.

First Term

The first term of an arithmetic sequence, denoted by \(a_1\), is the starting point from which all subsequent terms are generated. Finding \(a_1\) can sometimes require working backwards using both the sum formula and the nth-term formula.
In the provided exercise, the value of \(a_1\) was determined to be \(2\). This was calculated by combining information gained from the sum of the sequence and the expression for the 42nd term. Solving these steps indicated how each part of an arithmetic sequence interlinks. Finding the first term is pivotal for defining the entire arithmetic series and validating the relationship amongst its terms.