Problem 34 - Algebra and Trigonometry

Question

\(27-36\) . Determine the common difference, the fifth term, the nth term, and the 100 th term of the arithmetic sequence. $$ 15,12.3,9.6,6.9, \ldots $$

Step-by-Step Solution

Answer

Common difference: -2.7. Fifth term: 4.2. nth term: \(17.7 - 2.7n\). 100th term: -252.3.

Step 1: Identify the Common Difference

Let's determine the common difference of the arithmetic sequence. To do this, subtract the second term from the first term: \(12.3 - 15 = -2.7\). Thus, the common difference \(d\) is \(-2.7\).

Step 2: Calculate the Fifth Term

The terms in an arithmetic sequence are given by \(a_n = a + (n-1)d\), where \(a\) is the first term and \(d\) is the common difference. Now, calculate the fifth term: \(a_5 = 15 + (5-1)(-2.7) = 15 - 10.8 = 4.2\). Thus, the fifth term is 4.2.

Step 3: Find the General Formula for the nth Term

Using the formula for the nth term, \(a_n = a + (n-1)d\), substitute the known values \(a = 15\) and \(d = -2.7\): \(a_n = 15 + (n-1)(-2.7)\). Simplify to get \(a_n = 15 - 2.7n + 2.7\), which simplifies further to \(a_n = 17.7 - 2.7n\).

Step 4: Calculate the 100th Term

Substitute \(n = 100\) into the nth term formula: \(a_{100} = 17.7 - 2.7 \times 100 = 17.7 - 270 = -252.3\). Thus, the 100th term is -252.3.

Key Concepts

Common Difference

The common difference is a key feature of an arithmetic sequence. It’s the consistent amount added (or subtracted) to get from one term to the next in the sequence. To find the common difference, subtract the first term from the second.

• For the sequence: 15, 12.3, 9.6, 6.9, the common difference is calculated as:
  • \(12.3 - 15 = -2.7\)
• In this case, it is \(-2.7\).

A negative common difference shows that each subsequent term is decreasing by 2.7.

Nth Term Formula

The nth term formula of an arithmetic sequence helps you find any term in the sequence without having to list all the terms. The formula is expressed as:

• \(a_n = a + (n-1) \times d\)

where:

• \(a\) is the first term,
• \(d\) is the common difference,
• \(n\) is the term number.

In the given sequence, plugging the values gives:

• \(a_n = 15 + (n-1)(-2.7)\)
• This simplifies further to \(a_n = 17.7 - 2.7n\).

This formula allows you to calculate the value of any term by just substituting the desired term number \(n\).

Terms of a Sequence

Each number in an arithmetic sequence is called a term. The position of a term is crucial as it helps in both identifying and differentiating the terms within a sequence.

• The first term \(a\) of our sequence is 15.
• The fifth term, calculated using the formula, is 4.2.
• The 100th term is \(-252.3\).

These terms show how the sequence progresses by always integrating the common difference. When you know the general formula, you can quickly calculate any term within the sequence.

Arithmetic Progression

An arithmetic progression is simply another term for an arithmetic sequence. It highlights the regular step-like pattern of numbers, where each term after the first is derived by adding a constant (the common difference).

• The sequence: 15, 12.3, 9.6, 6.9 is a typical arithmetic progression.
• Each subsequent term is found by subtracting 2.7, which reflects our calculated common difference.

Understanding arithmetic progression opens up practical approaches to solving problems in real life, like predicting schedules, financial projections, and any scenario of uniform change over time.