Problem 27 - 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. $$a_{1}=9, d=2$$

Step-by-Step Solution

Answer

The formula for the general term of the given arithmetic sequence is \(a_n = 2n + 7\) and the 20th term, \(a_{20}\), is 47.

Step 1: Understanding Arithmetic Sequence General Term Formula

An arithmetic sequence is a sequence of numbers in which the difference of any two successive members is a constant. This constant is commonly denoted as \(d\) and it's called the common difference. For such a sequence, the nth term \(a_n\) can be described by the formula \(a_n = a_1 + (n-1)d\), where \(a_1\) is the first term and \(n\) is the term number.

Step 2: Substitute Known Values into the Formula

Given that the first term \(a_1\) of the sequence is 9 and the common difference \(d\) is 2, substitute these values into the formula for the nth term. Thus, the formula for the nth term \(a_n\) of the sequence becomes \(a_n = 9 + (n-1)2\).

Step 3: Simplify the General Formula

The formula can further be simplified by expanding the expression to arrive at \(a_n = 9 + 2n - 2\), which simplifies further to \(a_n = 2n + 7\). This is the formula for the general term of the given arithmetic sequence.

Step 4: Find the 20th Term of the Sequence

To find the 20th term of the sequence, \(a_{20}\), substitute \(n = 20\) into the general term formula \(a_n = 2n + 7\). This gives \(a_{20} = 2*20 + 7 = 47\).

Key Concepts

General Term Formula

The General Term Formula for an arithmetic sequence offers a way to calculate any term in the sequence without having to list out all the terms beforehand. This formula is expressed as: \[ a_n = a_1 + (n-1)d \] where:

• \( a_1 \) is the first term of the sequence.
• \( d \) is the common difference between consecutive terms.
• \( n \) is the position of the term within the sequence.

Knowing this formula is crucial, as it provides a direct path to find any term needed with minimal calculations. For instance, in the given problem, once the general term formula \( a_n = 2n + 7 \) is determined, any specific term can be easily calculated by substituting the appropriate value of \( n \).

Common Difference

The Common Difference is a key element of an arithmetic sequence. It is the constant amount that each term increases (or decreases) by as you move from one term to the next. You can calculate the common difference by subtracting any term from the term that directly follows it:

• \( d = a_{n+1} - a_n \)

In our example, the common difference \( d \) is 2. This means each term in the sequence adds 2 to the previous term. The common difference ensures that the sequence will always follow a consistent pattern, which is why the arithmetic sequence is so predictable.

Nth Term

The Nth Term represents the value of any term positioned at \( n \) within an arithmetic sequence. Unlike geometric sequences, the nth term in an arithmetic series does not require previous terms to calculate its value, thanks to the general term formula. Utilizing the formula:```plaintext a_n = 2n + 7```you would simply replace \( n \) with your term number to find its value. For the 20th term \( (n = 20) \), substitute to find:```plaintext a_{20} = 2*20 + 7 = 47```This ability to quickly compute the value for any term makes the nth term a powerful tool in solving arithmetic sequence problems efficiently.

Sequence

A Sequence in mathematics is an ordered set of numbers that follow a specific pattern or rule. In an arithmetic sequence, each term after the first is derived by adding a fixed, constant number called the "common difference". This structured progression ensures that the pattern remains uniform across all terms. For the sequence given in the problem, the numbers progress by consistently adding 2, starting from 9. Therefore, the sequence begins: ```plaintext 9, 11, 13, 15, ... ``` Understanding sequences help in visualizing and predicting future terms. With the provided values, we generate subsequent terms without much effort and directly apply the general term to derive any desired position in the sequence.