Problem 25 - College Algebra

Question

For the following exercises, find the specified term given two terms from an arithmetic sequence. $$ a_{3}=-17.1 \text { and } a_{10}=-15.7 . \text { Find } a_{21} $$.

Step-by-Step Solution

Answer

The 21st term, \(a_{21}\), is -13.5.

Step 1: Identify the Common Difference

In an arithmetic sequence, the difference between consecutive terms is constant, called the common difference (dd).). First, find \(d\) using \(a_3\) and \(a_{10}\).\[d = \frac{{a_{10} - a_{3}}}{{10 - 3}} = \frac{{-15.7 - (-17.1)}}{7} = \frac{1.4}{7} = 0.2\]

Step 2: Find the First Term

Use the formula for the nth term of an arithmetic sequence: \(a_n = a_1 + (n-1) imes d\). We know \(a_3 = -17.1\). Substitute the known values to solve for \(a_1\).\[-17.1 = a_1 + (3-1) imes 0.2\]\[-17.1 = a_1 + 0.4\]\[a_1 = -17.5\]

Step 3: Calculate the 21st Term

Now that we have \(a_1\) and \(d\), use the nth term formula again to find \(a_{21}\):\[a_{21} = a_1 + (21-1) imes d\]\[a_{21} = -17.5 + 20 imes 0.2\]\[a_{21} = -17.5 + 4\]\[a_{21} = -13.5\]

Key Concepts

Understanding the Common Difference

In every arithmetic sequence, a very key element is the **common difference**. This term refers to the consistent gap between any two consecutive numbers within the sequence. Knowing this difference helps in navigating through the sequence with ease. For example, in our problem, to find the common difference \( d \), we've considered the values of two terms: \( a_3 = -17.1 \) and \( a_{10} = -15.7 \).
To determine \( d \), subtract \( a_3 \) from \( a_{10} \), and then divide by the difference in their term numbers:

• Calculate the difference in values: \( -15.7 - (-17.1) = 1.4 \)
• Count the number of steps from the third term to the tenth term: \( 10 - 3 = 7 \)
• This gives us \( d = \frac{1.4}{7} = 0.2 \)

So, the common difference is \( 0.2 \). This constant helps to recognize how each term in the sequence stems from its preceding term. You simply add \( 0.2 \) to move forward in the sequence. Understanding and calculating this difference is foundational in working with arithmetic sequences.

Exploring the nth Term Formula

The **nth term formula** is a powerful tool that lets you find any term within the arithmetic sequence. It follows a simple formula: \[ a_n = a_1 + (n - 1) \times d \]Here, \( a_n \) denotes the nth term you wish to calculate, \( a_1 \) is the first term of the sequence, and \( d \) represents the common difference.
In our example, we utilized this formula to derive the sequence details. Initially, we didn't know \( a_1 \), the first term, so we rearranged our formula with the third term data:

• Substitute \( a_3 = -17.1 \) into the formula: \( -17.1 = a_1 + (3 - 1) \times 0.2 \)
• Calculate: \( -17.1 = a_1 + 0.4 \)
• Solve for \( a_1 \): \( a_1 = -17.5 \)

With this calculation, we've successfully identified the first element of the sequence. Utilizing this formula helps you find any position in the sequence with ease, just by plugging in the values.

Arithmetic Sequence Term Calculation

To calculate any term in an arithmetic sequence, once you know \( a_1 \) and \( d \), the nth term formula is all you need. For example, let’s find \( a_{21} \), the twenty-first term, using our already known values:

• First, we calculate the factor: \( (21-1) = 20 \).
• The formula then becomes: \( a_{21} = -17.5 + 20 \times 0.2 \).
• Perform the multiplication: \( 20 \times 0.2 = 4 \).
• Add to the first term: \( -17.5 + 4 = -13.5 \).

Thus, \( a_{21} = -13.5 \).
These calculations underscore the importance of understanding the formula and the common difference.
With these tools, finding any term becomes straightforward: just plug and solve! This method ensures you can navigate and understand any arithmetic sequence confidently.