Problem 36 - Precalculus: Mathematics for Calculus

Question

Determine the common difference, the fifth term, the \(n\) th term, and the 100 th term of the arithmetic sequence. $$64,49,34,19, \dots$$

Step-by-Step Solution

Answer

The common difference is -15, the fifth term is 4, the nth term formula is \(a_n = 79 - 15n \), and the 100th term is -1421.

Step 1: Identify the common difference

An arithmetic sequence follows a pattern where each term is obtained by adding a fixed number, known as the common difference, to the previous term. To find the common difference, subtract the first term from the second term: \( 49 - 64 = -15 \). Thus, the common difference is \(-15\).

Step 2: Calculate the fifth term

The formula for the fourth term can be applied to find the fifth term in the sequence as well. The fifth term can be found by adding the common difference to the fourth term: \(19 + (-15) = 4\). Therefore, the fifth term is \(4\).

Step 3: Determine the nth term formula

The nth term of an arithmetic sequence is given by the formula \( a_n = a_1 + (n-1) imes d \), where \(a_1\) is the first term and \(d\) is the common difference. Plug in the known values: \( a_n = 64 + (n-1)(-15) \). Simplify this to \( a_n = 64 - 15n + 15 \), and finally, \( a_n = 79 - 15n \).

Step 4: Calculate the 100th term

Using the formula for the nth term \( a_n = 79 - 15n \), you can find the 100th term by substituting \(n=100\): \( a_{100} = 79 - 15 imes 100 \). This simplifies to \( a_{100} = 79 - 1500 \), thus \( a_{100} = -1421 \).

Key Concepts

Common Difference

In an arithmetic sequence, each term after the first is found by adding a constant number known as the "common difference." This means that if you know any two consecutive terms in the sequence, you can easily determine this constant. For example, consider the terms 64 and 49 from the sequence you've given. To find the common difference, simply subtract the first term from the second:

• Formula: Common Difference (\( d \) ) = Second Term - First Term
• Here: \( d = 49 - 64 = -15 \)

So, the common difference is \( -15 \). It is negative, indicating that the sequence decreases as you move along. Understanding this concept is fundamental because it helps to shape the entire sequence.

nth Term

The "nth term" of an arithmetic sequence allows you to find any term in the sequence without listing all the terms. This is a powerful way to understand how sequences work. The formula to find the nth term is given by \( a_n = a_1 + (n-1) \times d \) . Here's what it means:

• \( a_n \): The nth term of the sequence.
• \( a_1 \): The first term of the sequence.
• \( d \): The common difference.
• \( n \): The position of the term in the sequence.

As an example, let's use your sequence: 64, 49, 34, 19...

• First term \( a_1 \) = 64
• Common difference \( d \) = -15

Let's plug these into the formula: \[ a_n = 64 + (n-1)(-15) \]This simplifies to \[ a_n = 79 - 15n \]. With this formula, you can now find any term in the sequence just by knowing its position.

Sequence Formula

The sequence formula in an arithmetic sequence is a great tool for predicting any term in the sequence. The general formula helps you calculate each term by incorporating both the first term and the common difference. Here's how it works:The formula is:

• \( a_n = a_1 + (n-1) \times d \)

Every arithmetic sequence is created using its first term and its common difference. Applying this formula:

• \( a_1 \) is typically the first term, which is 64 in the given sequence.
• \( d \) is the common difference of \(-15\).
• Replace with known values: \[ a_n = 64 + (n-1)(-15) \]
• This simplifies to \[ a_n = 79 - 15n \].

Having this formula at hand, you can quickly find any specific term, such as the 100th term in the exercise, leading to \( a_{100} = -1421 \). This formula connects all the elements of an arithmetic sequence harmoniously.