Problem 29 - College Algebra

Question

Write a formula for the general term (the nth term) of each arithmetic sequence. Do not use a recursion formula. Then use the formula for \(a_{n}\) to find \(a_{20}\), the 20 th term of the sequence. $$a_{1}=-20, d=-4$$

Step-by-Step Solution

Answer

The formula for the nth term of the arithmetic sequence is \(a_{n} = -4n - 16\) and the 20th term of the sequence is -96.

Step 1: Writing the General Term

The general term \(a_{n}\) of an arithmetic sequence can be written as \(a_{n} = a_{1} + (n - 1)*d\). Substituting the given values into this formula will give the general term of the sequence: \(a_{n} = -20 + (n - 1) * -4\). This is equivalent to \(a_{n} = -20 - 4n + 4\), and simplified further to \(a_{n} = -4n -16\).

Step 2: Calculating the 20th Term

Substitute n = 20 into the general formula \(a_{n} = -4n -16\), to find the 20th term: \(a_{20} = -4*20 - 16 = -80 - 16 = -96\).

Key Concepts

General Term

In an arithmetic sequence, the term formula is known as the "general term." It forms the backbone of understanding how a sequence progresses. The general term is given by the formula:

• \[ a_n = a_1 + (n - 1) imes d \]

Here, \( a_n \) is the nth term of the sequence, \( a_1 \) is the first term, \( n \) is the term number, and \( d \) is the common difference between consecutive terms. This formula allows us to find any term in the sequence without listing all the previous terms. For the sequence in the exercise, we have \( a_1 = -20 \) and \( d = -4 \). Substituting these values, the general term becomes \( a_n = -20 + (n - 1)(-4) \), which simplifies to \( a_n = -4n - 16 \). This expression will let us calculate any term's value directly.

nth Term

When we talk about the nth term in an arithmetic sequence, we're referring to any term in the sequence at a specified position \( n \). The formula for the nth term is the same as the general term's formula:

• \[ a_n = a_1 + (n - 1) imes d \]

This is an important concept because it allows students to determine a specific term's value quickly, especially in large sequences. To illustrate, in the exercise where we needed the 20th term, we used this formula to substitute \( n = 20 \) and found:

• \[ a_{20} = -4 imes 20 - 16 = -96. \]

This shows how efficiently the nth term formula works without needing recursion or additional repeated calculations.

Recursion Formula

A recursion formula, unlike the general term, expresses each term of a sequence based on the preceding one. While the exercise specifically asked not to use recursion, it's still useful to understand. A recursive formula for an arithmetic sequence looks like this:

• \[ a_n = a_{n-1} + d \]

This means each term can be derived from adding the common difference \( d \) to the previous term \( a_{n-1} \). Although this method can be simpler for calculating the next term, it becomes cumbersome for finding terms far from the start, as it requires knowing every preceding term.

Arithmetic Progression

An arithmetic progression, often called an arithmetic sequence, is a series of numbers where the difference between consecutive terms is constant. This constant is known as the "common difference," symbolized as \( d \). For example, with a first term \( a_1 = -20 \) and a common difference \( d = -4 \), the sequence begins:

• -20, -24, -28, -32,...

Each subsequent number in this series is created simply by adding \( d \) to the previous term. This kind of progression is foundational in mathematics because it demonstrates a simple, additive pattern. Recognizing it can make identifying patterns and calculating terms in sequences much more accessible.