Question

write a rule for the nth term of the sequence. Then fi nd a20. \(-2,-\frac{5}{4},-\frac{1}{2}, \frac{1}{4}, \ldots\)

Short answer

The 20th term of the sequence given by the rule is represented by the expression \(-2 \cdot (5/8)^{19}\)

Step-by-step solution

Identify if the sequence is Arithmetic or Geometric

Examine the sequence to determine if it is an arithmetic sequence (where the difference between each term is constant) or a geometric sequence (where the ratio of each successive pair of terms is constant). In this case, since the sequence is changing by common ratio, not by a common difference, it is a geometric sequence.

Find the Common Ratio

To find the common ratio in this sequence, divide any term by its previous term. For instance, -5/4 divided by -2 equals 5/8, or -1/2 divided by -5/4 also results in 5/8. The constant value that you get (5/8) is the common ratio.

Write the rule for the nth term

The nth term of a geometric sequence is given by the rule: \(a_n = a_1 \cdot r^{n-1}\), where \(a_n\) denotes the nth term, \(a_1\) is the first term, r is the common ratio and n is the position of the term in the sequence. However, in the given sequence the first term, \(a_1\) is -2 and the common ratio, r is 5/8. Hence the rule of the nth term will be \(a_n = -2 \cdot (5/8)^{n-1}\).

Find the 20th term of the sequence

Using the rule discovered in the previous step, replace 'n' with 20 to find the 20th term of the sequence. By plugging these values in, we get \(a_{20} = -2 \cdot (5/8)^{19}\). Remember not to simplify the exponent before multiplying, so avoid writing \(5^{19}/8^{19}\).

Key concepts

Common Ratio

In a geometric sequence, the common ratio is the key feature that distinguishes this type of sequence from an arithmetic sequence. The common ratio is the factor by which each term is multiplied to get the next term. It remains constant throughout the sequence.

To find the common ratio, you divide any term by its preceding term. For example, in the sequence \(-2, -\frac{5}{4}, -\frac{1}{2}, \frac{1}{4}, \ldots\), divide \(-\frac{5}{4}\) by \(-2\) to get \(\frac{5}{8}\). You can confirm this by also dividing \(-\frac{1}{2}\) by \(-\frac{5}{4}\), which will again give \(\frac{5}{8}\).

This constant ratio ensures that the pattern followed is consistent, allowing the calculation of any subsequent term using this single value.

Nth Term Formula

The nth term formula for a geometric sequence allows you to find any term's value without listing all terms first. This formula is expressed as \(a_n = a_1 \cdot r^{n-1}\) where:

• \(a_n\) is the nth term you wish to find.
• \(a_1\) is the first term of the sequence.
• \(r\) is the common ratio.
• \(n\) is the term number.

With this exercise, we have \(a_1 = -2\) and \(r = \frac{5}{8}\). Therefore, the formula becomes:\(a_n = -2 \cdot \left( \frac{5}{8} \right)^{n-1}\).

This understanding of how to create and use the formula simplifies the process of determining any term's value in the sequence.

Sequence Identification

Identifying the type of sequence is a vital step to determining how to approach solving it. There are mainly two types - arithmetic and geometric:

• An arithmetic sequence progresses by adding or subtracting a constant value to each term, known as the common difference.
• A geometric sequence progresses by multiplying by a constant value, known as the common ratio.

In the exercise provided, the sequence was recognized as geometric because the transformation from one term to the next involves a constant multiplication factor of \(\frac{5}{8}\).

Understanding the characteristics of each type of sequence helps you choose the correct method to find terms or analyze the behavior of the series. This foundation is crucial for solving any sequence-related problems effectively.