Problem 30 - Precalculus: Mathematical for Calculus

Question

Determine the common difference, the fifth term, the \(n\) th term, and the 100 th term of the arithmetic sequence. $$15,12.3,9.6,6.9, \dots$$

Step-by-Step Solution

Answer

Common difference is -2.7, fifth term is 4.2, general formula is \( a_n = 15 - 2.7(n-1) \), and 100th term is -252.3.

Step 1: Identify the Common Difference

An arithmetic sequence is one in which the difference between consecutive terms is constant. To find the common difference \( d \), subtract the second term from the first term: \[ d = 12.3 - 15 = -2.7 \] So, the common difference \( d = -2.7 \).

Step 2: Find the Fifth Term

Given that the first term \( a_1 = 15 \) and the common difference \( d = -2.7 \), the \( n \)-th term of an arithmetic sequence is given by:\[ a_n = a_1 + (n-1) imes d \]For the fifth term \( n=5 \):\[ a_5 = 15 + (5-1) imes (-2.7) \]\[ a_5 = 15 - 10.8 = 4.2 \]So, the fifth term is 4.2.

Step 3: Determine the General Formula for the \( n \)-th Term

The general formula for the \( n \)-th term (\( a_n \)) of an arithmetic sequence is:\[ a_n = a_1 + (n-1) imes d \]Substituting the known values, \( a_1 = 15 \) and \( d = -2.7 \):\[ a_n = 15 + (n-1) imes (-2.7) \]This simplifies to:\[ a_n = 15 - 2.7(n-1) \]

Step 4: Calculate the 100th Term

Using the formula derived for the \( n \)-th term, substitute \( n = 100 \):\[ a_{100} = 15 + (100-1) imes (-2.7) \]\[ a_{100} = 15 - 267.3 \]\[ a_{100} = -252.3 \]Therefore, the 100th term is -252.3.

Key Concepts

Common Difference

In an arithmetic sequence, every term after the first is generated by adding a constant value known as the "common difference" to the previous term. This means if you're given two consecutive terms, you can always find this difference by subtracting the first term from the second.

For example, consider the sequence provided: 15, 12.3, 9.6, and so on. By finding the common difference of this sequence:

• Subtract the second term from the first: 12.3 - 15.
• The result is -2.7, which is the common difference (d) in this case.

When a sequence of numbers consistently decreases by the same amount (as it does here), this negative difference (-2.7) shows that the sequence is decreasing.
The common difference is crucial for determining any term in the sequence and forms the basis of constructing the general formula for the sequence.

General Formula for n-th Term

To find a specific term in an arithmetic sequence without listing all preceding terms, you can use the general formula for the \( n \)-th term. This powerful formula allows you to calculate directly any term in the sequence based on its position \( n \).

It is given by:

• \( a_n = a_1 + (n - 1) \times d \)

Where:

• \( a_n \) is the \( n \)-th term we want to find.
• \( a_1 \) is the first term of the sequence.
• \( (n - 1) \) is the number of intervals from the first term to the \( n \)-th term.
• \( d \) is the common difference.

By substituting the known values from our sequence, first term \( a_1 = 15 \) and common difference \( d = -2.7 \), we get:

• \( a_n = 15 + (n-1) \times (-2.7) \)
• This equation is the general formula you can use for any term \( n \) in this sequence.

It effectively opens up the capacity to calculate terms like the 100th term, without manually going through all previous terms.

100th Term of Sequence

To find the 100th term of an arithmetic sequence, employ the general formula for the \( n \)-th term. We have already derived this formula as:

• \( a_n = 15 + (n-1) \times (-2.7) \)

We simply substitute \( n = 100 \) into this equation:

• \( a_{100} = 15 + (100 - 1) \times (-2.7) \)

This simplifies to:

• \( a_{100} = 15 + 99 \times (-2.7) \)
• Calculate \( 99 \times -2.7 = -267.3 \)
• Finally, \( 15 - 267.3 = -252.3 \)

Thus, the 100th term in the sequence is \(-252.3\).
This step illustrates how using the general formula can save a tremendous amount of time and effort for finding terms far along in a sequence. It is a prime example of how arithmetic sequences simplify otherwise tedious calculations.