Problem 66 - Algebra 2

Question

Find the 27 th term of each sequence. $$ 2.1,1.7,1.3, \dots $$

Step-by-Step Solution

Answer

The 27th term of the sequence is -8.3

Step 1: Identify the first term, common difference, and term number

The first term, \(a_1 = 2.1\), the common difference, \(d = -0.4\), and the term number, \(n = 27\).

Step 2: Substitute the values in the formula

Substitute \(a_1 = 2.1\), \(d = -0.4\), and \(n = 27\) in the formula \(a_n = a_1 + (n - 1) * d\). So, the formula becomes \(a_{27} = 2.1 + (27 - 1) * (-0.4)\).

Step 3: Calculation

Calculate the right-hand side of the equation from Step 2. Therefore, \(a_{27} = 2.1 + 26 * (-0.4) = 2.1 -10.4 = -8.3\)

Step 4: Conclusion

The 27th term of the sequence is -8.3

Key Concepts

Common Difference

In any arithmetic sequence, the common difference is a crucial value. It determines how much each term differs from the previous one. This difference remains constant throughout the sequence. For instance, if you look at the sequence 2.1, 1.7, 1.3, you will notice a pattern. You can find the common difference by subtracting the first term from the second term:

• Subtract 1.7 – 2.1 = -0.4.
• The common difference is -0.4.

This information will help you predict future terms or determine past terms in the sequence. Understanding the common difference is essential in navigating arithmetic sequences efficiently.

Formula for nth term

To find a specific term in an arithmetic sequence, use the formula:
\[ a_n = a_1 + (n - 1) imes d \]This formula allows you to calculate the nth term (the term number you want) easily. In this expression:

• \( a_n \) is the nth term you are trying to find.
• \( a_1 \) is the first term of the sequence.
• \( n \) is the term number you need.
• \( d \) is the common difference.

For instance, if you are trying to find the 27th term of a sequence where the first term is 2.1 and the common difference is -0.4, you plug these into the formula.
Understanding and utilizing this formula will help you solve many arithmetic sequence problems effectively.

Sequence Term

A sequence term is a specific term position in a sequence. When dealing with arithmetic sequences, each sequence term is derived by adding the common difference repeatedly starting from the first term. In our exercise, we identified the first term as 2.1.

• The 2nd term is 1.7.
• The 3rd term is 1.3.

To find the 27th term, you don't need to calculate each successive term. Instead, use the formula for the nth term. Remember, every term in the sequence is dependent on both the first term and the common difference.

Negative Common Difference

What happens when the common difference is negative? A negative common difference means each term in the sequence is smaller than the one before it. Consider our sequence: 2.1, 1.7, and 1.3. The sequence decreases because the common difference, -0.4, is negative.

• This indicates the sequence is going downwards as it progresses.
• The terms become increasingly smaller.

This knowledge is vital in anticipating the overall direction and trend of the sequence. When dealing with real-world scenarios or more complex mathematical problems, understanding the impact of a negative common difference can greatly enhance your analytical skills.