Problem 30 - A Graphical Approach to College Algebra

Question

Find \(a_{1}\) for each arithmetic sequence. $$S_{20}=-1300, a_{20}=-122$$

Step-by-Step Solution

Answer

The first term \(a_1\) is \(-8\).

Step 1: Understanding the Problem

The goal is to find the first term \(a_1\) of an arithmetic sequence. We are given the sum of the first 20 terms \(S_{20} = -1300\) and the 20th term \(a_{20} = -122\).

Step 2: Formula for the Sum of an Arithmetic Sequence

The formula for the sum of the first \(n\) terms of an arithmetic sequence is \(S_n = \frac{n}{2} (a_1 + a_n)\). For this problem, \(n = 20\), \(S_{20} = -1300\), and \(a_{20} = -122\).

Step 3: Substitute Known Values

Substitute the known values into the formula: \(-1300 = \frac{20}{2} (a_1 + (-122))\).

Step 4: Simplify the Sum Formula

Simplify the equation: \(-1300 = 10(a_1 - 122)\).

Step 5: Solve for \(a_1\)

Divide both sides of the equation by 10: \(-130 = a_1 - 122\). Then solve for \(a_1\) by adding 122 to both sides: \(a_1 = -130 + 122 = -8\).

Key Concepts

Sum of Arithmetic Sequence

The sum of an arithmetic sequence can be calculated using a specific formula that gathers all the sequence terms together. Understanding this formula helps us determine how the terms of a sequence accumulate over time. In arithmetic sequences, every term is derived by adding a constant difference to the previous term. If you need the sum of the whole sequence up to a certain point, this can be easily calculated.

The formula for the sum of the first \(n\) terms in an arithmetic sequence is:

• \(S_n = \frac{n}{2} (a_1 + a_n)\)

Here, \(S_n\) is the sum of the first \(n\) terms, \(n\) is the number of terms, \(a_1\) is the first term, and \(a_n\) is the last term in the section of the sequence you are summing up. This formula works because it essentially averages the first and the last term, then multiplies by the number of terms.

Sequence Terms

In an arithmetic sequence, the terms follow a specific order characterized by a common difference between consecutive terms. This common difference is a fixed amount added to each term to arrive at the next term.

For example, if you start with the term \(a_1\), to find the next term \(a_2\), you add the common difference \(d\). This means:

• \(a_2 = a_1 + d\)
• \(a_3 = a_2 + d\)
• ... and so on.

This pattern allows you to predict any term in the sequence given the first term and the common difference. Each term can be noted as \(a_n = a_1 + (n-1)d\), which is crucial for finding specific values like the 20th term, \(a_{20}\), in this exercise.

Formula for Arithmetic Sequence

The formula for arithmetic sequences provides a way to find any term in the sequence without having to manually add the common difference each time. This formula is incredibly useful in both academic and real-world contexts.

• General Formula: \(a_n = a_1 + (n-1)d\)

This formula helps you calculate the value of any term by knowing the first term \(a_1\) and the common difference \(d\). Suppose you want to determine \(a_{20}\), and you already know \(a_1\) and \(d\). You simply plug these into the equation, ensuring you solve for \(a_n\) with context:

• For finding the common difference, rearrange: \(d = \frac{a_{20} - a_1}{19}\)

In this specific exercise, knowing \(a_{20} = -122\) and \(S_{20} = -1300\) enables us to utilize formulas efficiently to solve for unknowns like \(a_1\)."