Question

Describe the pattern, write the next term, and write a rule for the \(\boldsymbol{n}\) th term of the sequence. \(5.8,4.2,2.6,1,-0.6 \ldots\)

Short answer

The pattern is a decrease by 1.6 for each term. The next term in the sequence is -2.2. The nth term of the sequence is represented by the rule \(5.8 + (n-1)(-1.6)\).

Step-by-step solution

Identify the Pattern

The second term (4.2) subtracted from the first term (5.8) yields -1.6. The third term (2.6) subtracted from the second term (4.2) results in the same difference, -1.6. Since all the consecutive differences are the same, determine that the sequence is a linear (arithmetic) sequence.

Find the Next Term

The next term can be found by subtracting the common difference from the last term given in the sequence. Subtract -1.6 from -0.6 yields -2.2 as the next term.

Derive the nth Term Rule

The general formula for an arithmetic sequence is \(a_n = a_1 +(n-1)d\), where \(a_n\) is the nth term, \(a_1\) is the 1st term of the sequence and \(d\) is the common difference. Therefore, the nth term can be expressed as \(5.8 + (n-1)(-1.6)\).

Key concepts

Pattern Recognition

Recognizing patterns in a sequence is the first and crucial step toward understanding and describing the nature of the sequence. A pattern can be described as a noticeable regularity in the sequence of numbers. In the exercise provided, the difference between consecutive terms is constant, indicating an arithmetic sequence.

Students should practice observing this consistent difference by subtracting adjacent terms: from 5.8 to 4.2, the difference is -1.6, and we notice the same difference again when we subtract 4.2 from 2.6. This pattern of consistent difference is what defines an arithmetic sequence, and once you have identified this, you've paved the way to solving the problem fully.

To enhance understanding, it's beneficial to look at various sequences and determine if a pattern exists. This can involve recognizing sequences that continually add or subtract the same number, or even multiply or divide by a fixed number, which would indicate geometric sequences instead of arithmetic.

nth Term

The nth term of a sequence refers to an algebraic expression that allows you to find the value of any term in the sequence without listing all the terms. It's a formula that describes the general relationship between the term number and its value. For arithmetic sequences, the nth term is given by the formula \(a_n = a_1 + (n-1)d\), where \(a_n\) is the nth term you’re looking to find, \(a_1\) is the first term of the sequence, and \(d\) is the common difference between consecutive terms.

Using this formula can help students not only find any term in a sequence without having to write out the whole sequence but also understand the structure of the sequence at a deeper level. For the given sequence, by realizing that the common difference \(d\) is -1.6, students can formulate the nth term as \(5.8 + (n-1)(-1.6)\). This expression now allows anyone to calculate the value of any term merely by plugging in different values of n.

Linear Sequence

A linear sequence, commonly known as an arithmetic sequence, has a constant difference between consecutive terms and represents a linear pattern in mathematics. The relationship between the terms is additive (or subtractive, by a constant amount). In the given exercise, by identifying that the sequence decreases by 1.6 each time, students can understand and predict the behavior of the sequence.

Arithmetic sequences are linear because, when plotted on a graph with the term number along the x-axis and the term value on the y-axis, the points lie on a straight line. This linear nature makes arithmetic sequences simple to work with and forms a foundational concept for more complex mathematical ideas like arithmetic progressions and algebraic functions.