Problem 35 - 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. $$29,11,-7,-25 \dots$$

Step-by-Step Solution

Answer

Common difference = -18, 5th term = -43, \(n\)th term = \(47 - 18n\), 100th term = -1753.

Step 1: Identify the Common Difference

First, identify the common difference of the sequence. The common difference, denoted by \(d\), is the difference between consecutive terms in the sequence. Calculate \(d\) as the difference between the second term and the first term: \[ d = 11 - 29 = -18 \] Verify it contributes consistently to the sequence: \( -7 - 11 = -18 \) and \( -25 - (-7) = -18 \).

Step 2: Calculate the Fifth Term

Use the formula for the \(n\)th term of an arithmetic sequence: \( a_n = a_1 + (n-1) imes d \). Here, \( a_1 = 29 \), \( d = -18 \), and \( n = 5 \):\[ a_5 = 29 + (5-1) imes (-18) \] \[ a_5 = 29 + 4 imes (-18) \] \[ a_5 = 29 - 72 \] \[ a_5 = -43 \] Thus, the fifth term is \(-43\).

Step 3: Determine the General Term Formula

Develop the formula for the \(n\)th term of the sequence. The sequence begins with \( a_1 = 29 \) and has a common difference of \(d = -18\). The formula is given by:\[ a_n = a_1 + (n-1) imes d \] \[ a_n = 29 + (n-1) imes (-18) \] Simplifying gives:\[ a_n = 29 - 18n + 18 \] \[ a_n = 47 - 18n \]

Step 4: Calculate the 100th Term

Using the \(n\)th term formula, substitute \(n = 100\) to find the 100th term:\[ a_{100} = 47 - 18 imes 100 \] \[ a_{100} = 47 - 1800 \] \[ a_{100} = -1753 \] Thus, the 100th term of the sequence is \(-1753\).

Key Concepts

Common Difference

The common difference is a key element in understanding arithmetic sequences. It tells you how much each term in the sequence increases or decreases compared to the previous term. You find it by subtracting any term from the one that follows it. In the sequence provided: 29, 11, -7, -25, the common difference, denoted as \(d\), can be calculated as follows:

• Subtract the first term from the second term: \(11 - 29 = -18\).
• Verify this result with other consecutive terms: \(-7 - 11 = -18\) and \(-25 - (-7) = -18\).

This confirms that the sequence decreases by 18 each time, giving us consistency and structure to our progression. Understanding the common difference is crucial as it forms the basic building block for predicting future terms in the sequence.

Nth Term Formula

The nth term formula allows us to find an expression for any term in an arithmetic sequence, without having to list all previous terms. It's a simple yet powerful equation:\[ a_n = a_1 + (n-1) \times d \]Where:

• \(a_n\) is the term we want to find.
• \(a_1\) is the first term in the sequence.
• \(d\) is the common difference.
• \(n\) is the term number.

In our sequence, the first term \(a_1\) is 29, and the common difference \(d\) is -18. Placing these values into our formula gives us:\[ a_n = 29 + (n-1) \times (-18) \]Simplifying this, we obtain:\[ a_n = 47 - 18n \]This general formula allows anyone to calculate any term in the sequence quickly and efficiently by simply plugging in the desired term number \(n\).

Arithmetic Progression

An arithmetic progression (AP) is a sequence of numbers in which the difference of any two successive members is a constant, known as the common difference. This type of sequence is linear, meaning it forms a straight line when graphed. For example, in our sequence 29, 11, -7, -25:

• Each term is derived from adding or subtracting the common difference \(-18\).
• The sequence decreases in consistent steps, showcasing the arithmetic progression.

An arithmetic progression is important in both mathematics and daily life. It helps solve various problems where quantity changes at a constant rate. Whether you are adding savings in a bank or calculating scores over a period, arithmetic progressions can simplify complex analysis into manageable counts.