Problem 28 - Algebra and Trigonometry

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. $$ a_{1}=6, d=3 $$

Step-by-Step Solution

Answer

The formula for the nth term of the sequence is \(a_{n} = 3n + 3\), and the 20th term of the sequence is 63.

Step 1: Understanding the formula for the nth term of an arithmetic sequence

The nth term (denoted as \(a_n\)) of an arithmetic sequence can be found using the formula: \(a_{n} = a_{1} + (n-1) \cdot d\), where \(a_{1}\) is the first term, n is the term number and d is the common difference.

Step 2: Insert the given values into the formula

We need to replace \(a_{1}\) with 6 and d with 3 in the formula to obtain the general formula for the nth term: \(a_{n} = 6 + (n-1) \cdot 3\)

Step 3: Simplify the formula

Simplify to get the formula for the nth term: \(a_{n} = 3n + 3\)

Step 4: Find the 20th term

Now to find the 20th term, replace 'n' with 20 in the formula: \(a_{20} = 3 \cdot 20 + 3 = 60 + 3 = 63\)

Key Concepts

General Term Formula

An important aspect of arithmetic sequences is understanding the concept of the general term formula, also known as the nth term formula. This formula helps us find any term in the sequence without having to list all the preceding terms.
At the core, the general term formula for an arithmetic sequence is:

• \( a_{n} = a_{1} + (n-1) \cdot d \)

Here:

• \( a_{n} \) is the nth term you are trying to find.
• \( a_{1} \) represents the first term of the sequence.
• \( n \) is the position number of the term in the sequence.
• \( d \) is the common difference, the amount each term increases by from the last term.

This formula allows you to quickly calculate any term in the sequence, saving time and effort by not having to find all the previous terms first.

Common Difference

The common difference is a key feature of arithmetic sequences and plays a crucial role in determining the progression of numbers in the series. It is the consistent interval by which each term in the sequence increases or decreases compared to the previous term.
To calculate the common difference \( d \), you subtract any term in the sequence from the term that follows it:

• \( d = a_{n+1} - a_{n} \)

For the arithmetic sequence given in the exercise, with a first term \( a_{1} = 6 \) and common difference \( d = 3 \), each term grows by adding 3 to the previous one, like 6, 9, 12, and so on. Recognizing the common difference allows us to build or extend the sequence easily and is essential for using the general term formula.

Nth Term

In an arithmetic sequence, the nth term refers to the term located at a specific position within the sequence. Using the nth term formula, you can directly calculate any term's value without listing all the preceding terms or performing iterative calculations.
The formula for finding the nth term of an arithmetic sequence is given by:

• \( a_{n} = a_{1} + (n-1) \cdot d \)

For example, in the exercise, the nth term formula becomes \( a_{n} = 3n + 3 \) after inserting the given values. This simplification means that for any term \( n \), you can substitute \( n \) directly into the formula to find its value. By applying this to the 20th term, substituting \( n = 20 \) yields \( a_{20} = 63 \), confirming its position in the sequence.

Sequence Formula

A sequence formula is a mathematical expression that defines the pattern or rule of a number sequence. For arithmetic sequences, this formula provides a straightforward way to calculate any term in the sequence.
The general form of the sequence formula for arithmetic sequences is:

• \( a_{n} = a_{1} + (n-1) \cdot d \)

By using this sequence formula, you can establish a quick method to derive any number within the arithmetic sequence, whether it's the next term after a given number or the 100th term far along in the sequence. The sequence formula packages all necessary details about the sequence's rate of progression through the common difference and the starting point, \( a_{1} \). This makes solving problems involving arithmetic sequences both efficient and clear.