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Chapter 13: Problem 22

Determine an expression for the general term of each arithmetic sequence. $$ a_{1}=5, d=3 $$

Short Answer

Expert verified
\(a_n = 8 \left( -\frac{1}{4} \right)^{n-1}\)

Step by step solution

01

- Identify the First Term

The first term of the geometric sequence is given. Let's denote it by \(a_1\). Here, \(a_1 = 8\).
02

- Determine the Common Ratio

The common ratio \(r\) can be found by dividing the second term by the first term. Thus, \(r = \frac{-2}{8} = -\frac{1}{4}\). To confirm \(r\), check the next term: \(r = \frac{1/2}{-2} = -\frac{1}{4}\). The common ratio is consistent.
03

- Write the General Form of the nth Term

The general term of a geometric sequence is given by the formula: \(a_n = a_1 \, r^{n-1}\).
04

- Substitute the Known Values

Substitute the first term \(a_1 = 8\) and the common ratio \(r = -\frac{1}{4}\) into the general formula: \(a_n = 8 \left( -\frac{1}{4} \right)^{n-1}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

General Term of a Geometric Sequence
Understanding the general term of a geometric sequence helps determine any term in the sequence without listing all previous terms. A geometric sequence is defined by its first term and common ratio. The general term formula is key in identifying the value of the sequence at any position.
For any geometric sequence, the general term can be written as:
\[a_n = a_1 \times r^{(n-1)}\]
Here,
  • a_n is the term you want to find,
  • a_1 is the first term,
  • and r is the common ratio.
Utilizing the formula requires knowing the first term and the common ratio. This allows us to derive any term from the sequence directly.
For example, given the sequence 8, -2, \( \frac{1}{2} \), ..., the general term is determined as:
\[a_n = 8 \left( -\frac{1}{4} \right)^{(n-1)}\]
Common Ratio
The common ratio in a geometric sequence is the constant factor between consecutive terms. It determines the sequence鈥檚 behavior, whether it grows, shrinks, or alternates.
To find the common ratio, divide the second term by the first term:\[r = \frac{a_2}{a_1}\]
Checking consistency is crucial by applying the ratio to other pairs of terms. For instance, in our example:\[r = \frac{-2}{8} = -\frac{1}{4}\]
Validate with another pair of terms:\[r = \frac{1/2}{-2} = -\frac{1}{4}\]
This consistency confirms that the common ratio is \( -\frac{1}{4} \).
Knowing the common ratio is essential for fully understanding the sequence and finding its general term.
First Term
Identifying the first term is the simplest step in generating a geometric sequence. The first term is the starting point and is key to constructing the rest of the sequence.
In our example, the first term is denoted as \(a_1\). Here, \(a_1 = 8\).
Once the first term is known, along with the common ratio, the entire sequence can be described.
It is worth remembering that the first term sets the foundation for the subsequent terms through multiples of the common ratio:
  • Second term : \(a_1 \times r\)
  • Third term : \(a_1 \times r^2\)
For this sequence, the terms are generated starting from 8 and using the common ratio \( -\frac{1}{4}\):
  • First term \(a_1 = 8\)
  • Second term \(a_2 = 8 \left( -\frac{1}{4} \right) = -2\)
  • Third term \(a_3 = 8 \left( -\frac{1}{4} \right)^2 = \frac{1}{2}\)

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