Problem 65 - Algebra 2

Question

Find the 27 th term of each sequence. $$ 1,3.5,6,8.5, \dots $$

Step-by-Step Solution

Answer

The 27th term of the given arithmetic sequence is 66.

Step 1: Identify the Common Difference

The given sequence is 1, 3.5, 6, 8.5, and so on. By subtracting the first term from the second, and the second from the third, a common difference of 2.5 is discovered.

Step 2: Use the Formula for the General Term

The formula for the nth term of an arithmetic sequence is given by \(a_n = a_1 + (n-1)d\), where \(a_n\) is the nth term, \(a_1\) is the first term, d is the common difference, and n is the position of the term in the sequence.

Step 3: Insert Values into the Formula

We can use the formula, substituting the known values into it. Hence, \( a_27 = 1 + (27-1) * 2.5 \)

Step 4: Calculate

Finally, we can calculate the value. It gives \( a_27 = 1 + 26 * 2.5 = 1 + 65 = 66 \)

Key Concepts

Common Difference

In an arithmetic sequence, the common difference is a key component that defines the sequence. It is the consistent amount added (or subtracted) from one term to the next to generate the sequence. In the sequence given in the exercise: 1, 3.5, 6, 8.5, ..., you can find the common difference by subtracting the first term from the second term, and the second term from the third term, and so on. For example:

• 3.5 - 1 = 2.5
• 6 - 3.5 = 2.5
• 8.5 - 6 = 2.5

This arithmetic difference of 2.5 is consistent between all terms, confirming it to be the common difference. Understanding the common difference is crucial as it allows us to predict future terms in a sequence.

General Term Formula

The general term formula is the mathematical expression that represents the nth term of an arithmetic sequence. It is expressed as:\[ a_n = a_1 + (n-1)d \]Where:

• \( a_n \) is the nth term we are trying to find
• \( a_1 \) is the first term of the sequence
• \( d \) is the common difference
• \( n \) is the term number or position in the sequence

Using this formula, you can determine any term in an arithmetic sequence without listing all previous terms. This formula is essential because it allows you to quickly find terms even if they are far into the sequence.

nth Term Calculation

To calculate the nth term in the sequence, you apply the values you already know into the general term formula. Let's use our sequence example here, where we need to find the 27th term.Substitute the known values into the formula:

• \( a_1 = 1 \) (the first term)
• \( d = 2.5 \) (the common difference)
• \( n = 27 \) (the term number we are calculating)

Applying these values to the general term formula:\[ a_n = 1 + (27-1) \times 2.5 \]Therefore, \( a_{27} = 1 + 26 \times 2.5 \), which simplifies to:\[ a_{27} = 1 + 65 = 66 \]This calculation shows the 27th term of the sequence is 66. Understanding this process is valuable for using arithmetic sequences in real-world scenarios, like finance and statistics.