Problem 10 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 3-10: Find the \(n\)th term, the fifth term, and the tenth term of the arithmetic sequence. $$ \log 1000, \log 100, \log 10, \log 1, \ldots $$

Step-by-Step Solution

Answer

The nth term is \(4-n\); 5th term is -1; 10th term is -6.

Step 1: Understand the Sequence

The given sequence is: \( \log 1000, \log 100, \log 10, \log 1, \ldots \). Rewrite the logarithms in base 10 as whole numbers: \( \log 1000 = 3 \), \( \log 100 = 2 \), \( \log 10 = 1 \), and \( \log 1 = 0 \). This reveals that the sequence is \( 3, 2, 1, 0, \ldots \).

Step 2: Establish the Common Difference

An arithmetic sequence has a common difference \( d \) between consecutive terms. Here, \( d = 2 - 3 = -1 \). The sequence is \( 3, 2, 1, 0, \ldots \) with a common difference of \( -1 \).

Step 3: Formulate the General Term

The general formula for the \( n \)th term of an arithmetic sequence is \( a_n = a_1 + (n-1)d \). For this sequence, the first term \( a_1 = 3 \) and the common difference \( d = -1 \). Substitute these into the formula to get \( a_n = 3 + (n-1)(-1) \). Simplify to \( a_n = 3 - n + 1 = 4 - n \).

Step 4: Find the 5th Term

Using the formula \( a_n = 4 - n \), substitute \( n = 5 \) to find the 5th term: \( a_5 = 4 - 5 = -1 \).

Step 5: Find the 10th Term

Using the same formula, substitute \( n = 10 \) to find the 10th term: \( a_{10} = 4 - 10 = -6 \).

Key Concepts

nth term formula

The nth term formula is essential when dealing with arithmetic sequences. It allows us to find any term in the sequence without having to list all the previous terms. This is particularly helpful when looking for terms further down the sequence.

For an arithmetic sequence, the nth term is given by the formula:

• \( a_n = a_1 + (n-1) \, d \)

Where:

• \( a_n \) is the nth term,
• \( a_1 \) is the first term, and
• \( d \) is the common difference between consecutive terms.

In our example, we identified the first term \( a_1 \) as 3, and \( d \), the common difference, as -1. Plugging these values into the formula, we derived:

• \( a_n = 4 - n \)

This simplifies finding any term in the sequence without recalculating entire series.

common difference

The common difference is the consistent difference between consecutive terms in an arithmetic sequence. This property is what makes a sequence 'arithmetic.'

In our exercise, the sequence began as \( \, \log 1000, \log 100, \log 10, \log 1, \ldots \, \). When translated to whole numbers: 3, 2, 1, 0, the common difference was computed as follows:

• \( d = 2 - 3 = -1 \)

This -1 means each term is 1 less than the term before it. Recognizing the common difference helps us set up the nth term formula, ensuring accuracy and providing insight into the sequence's pattern.

logarithmic sequence

A sequence of logarithms, like the one we started with, is often transformed for easier manipulation, as was done here. The initial sequence given was \( \, \log 1000, \log 100, \log 10, \log 1, \ldots \, \).

• First, express each logarithm: \( \log_{10} 1000 = 3 \), \( \log_{10} 100 = 2 \), \( \log_{10} 10 = 1 \), and \( \log_{10} 1 = 0 \).
• This conversion changes the sequence to simple integers: 3, 2, 1, 0.

Understanding logarithmic sequences is crucial since they often reduce to arithmetic sequences, making subsequent calculations more straightforward. By recognizing patterns in logarithms, you can simplify and solve problems efficiently.

step-by-step solution

A detailed approach is sometimes the best way to truly understand how to work through exercises. Let's break down our sequence problem again:

1. **Understand the Sequence**: Converted each logarithm to a number: \( \log 1000 = 3 \), \( \log 100 = 2 \), etc.2. **Identify the Common Difference**: Calculated as -1 (e.g., \( 2 - 3 \)).3. **Formulate the nth Term**: Used the formula \( a_n = 4 - n \).4. **Calculate Specific Terms**: - Fifth Term: \( a_5 = 4 - 5 = -1 \) - Tenth Term: \( a_{10} = 4 - 10 = -6 \)Breaking the problem into manageable pieces allows anyone to grasp even a difficult concept by tackling each section methodically. This step-by-step breakdown ensures you leave no stone unturned, whether finding terms or understanding the sequence's structure.