Question

write a rule for the nth term of the sequence. Then fi nd a20. \(2.3,1.5,0.7,-0.1, \ldots\)

Short answer

The rule for the nth term of the sequence is \(a = 2.3 + (n - 1) * -0.8\), and the 20th term of the sequence (a20) is -12.9.

Step-by-step solution

Identify the Pattern

Consider the sequence given \(2.3, 1.5, 0.7,-0.1, \ldots\). You might notice that it decreases by a constant difference. We can find the difference between any two consecutive terms (subtracting the first term from the second, the second term from the third, and so on) to get a constant difference of \(-0.8\). Thus, this sequence is an arithmetic sequence.

Write a Rule for the nth Term

In an arithmetic sequence, we can calculate the nth term using the formula: first term + (n - 1) * difference. Here, the first term is 2.3 and the difference is -0.8. Hence, the general term a (where a is the nth term) of the sequence can be written as \(a = 2.3 + (n - 1) * -0.8\).

Find a20

Now, to find the 20th term of the sequence (a20), replace n in the formula with 20 to get \(a_{20} = 2.3 + (20 - 1) * -0.8\). Calculate this expression to get the 20th term. Thus, \(a_{20} = 2.3 -(19*0.8) = 2.3 - 15.2 = -12.9\).

Key concepts

Understanding the nth Term of a Sequence

Learning how to determine the nth term of a sequence is crucial for analyzing patterns and predicting future values in a series of numbers. This concept is most commonly used in arithmetic sequences, where the difference between consecutive terms is constant.

For a sequence such as the one mentioned, \(2.3, 1.5, 0.7, -0.1, \ldots\), the key is to first identify the common difference, which can be found by subtracting any term from its subsequent term. In this case, the common difference is \( -0.8 \). With this information, you can write a general rule to find the nth term: \( a_n = a_1 + (n - 1)d \), where \( a_1 \) is the first term, \(n\) is the term's position in the sequence, and \(d\) is the common difference.

In other words, each term is generated by starting with the first term and adding the common difference multiplied by one less than the term's position in the sequence. This enables us to calculate not only the upcoming terms but any term at any position, which is exceptionally useful when dealing with long sequences or seeking a specific term far down the sequence without having to calculate all previous terms.

Summing Up an Arithmetic Series

An arithmetic series is the sum of the terms in an arithmetic sequence. If you're asked to find the sum of the first 20 terms of our sequence (or any other number of terms), there's an efficient formula you can use, bypassing the need to add each term one by one.

The formula for the sum of an arithmetic series, \( S_n \), is \( S_n = \frac{n}{2}(a_1 + a_n) \) or \( S_n = \frac{n}{2}(2a_1 + (n-1)d) \) where \( n \) is the number of terms, \( a_1 \) is the first term, and \( a_n \) is the nth term. The second version of the formula is beneficial when the nth term is not known beforehand. This formula comes from the idea that the sum of an arithmetic series is the average of the first and last term, multiplied by the number of terms.

Defining Arithmetic Progression

An arithmetic progression is a sequence of numbers in which each term after the first is obtained by adding a constant, known as the common difference, to the previous term. Mathematically, this is represented as \( a_n = a_1 + (n - 1)d \).

It is this predictability and uniformity of increase or decrease that defines an arithmetic progression. In educational curriculum, understanding arithmetic progressions serves as a foundation for more complex mathematical concepts and real-world applications, such as financial calculations involving interest rates, and is essential for students who seek to excel in mathematical disciplines.

Each term of the progression is an element of the pattern, and the entire sequence embodies the concept of predictability in mathematics. This helps students to not only work with numbers in a theoretical framework but also to recognize such patterns in practical scenarios and datasets.