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Chapter 12: Problem 11

Find a general term for each geometric sequence. $$ -2, \frac{2}{3},-\frac{2}{9}, \ldots $$

Short Answer

Expert verified
The general term is \( -2 \times \bigg(-\frac{1}{3}\bigg)^{(n-1)} \).

Step by step solution

01

Identify the first term

The first term of the geometric sequence is often denoted by the symbol \( a \). From the given sequence, the first term \( a \) is \( -2 \).
02

Determine the common ratio

To find the common ratio \( r \), divide the second term by the first term or the third term by the second term: \[ r = \frac{\frac{2}{3}}{-2} = -\frac{1}{3} \. \] Verify by dividing the third term by the second term: \[ r = \frac{-\frac{2}{9}}{\frac{2}{3}} = -\frac{1}{3} \. \] Thus, the common ratio \( r \) is \( -\frac{1}{3} \).
03

Write the general formula for the geometric sequence

The general term of a geometric sequence can be written as \( a_n = a \times r^{(n-1)} \), where \( a_n \) is the \( n \)-th term, \( a \) is the first term, and \( r \) is the common ratio.
04

Substitute the values into the formula

Substitute \( a = -2 \) and \( r = -\frac{1}{3} \) into the general formula: \[ a_n = -2 \times \bigg(-\frac{1}{3}\bigg)^{(n-1)} \. \] This formula gives the general term of the geometric sequence.

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Key Concepts

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

common ratio
In any geometric sequence, the common ratio is a crucial element. It is the factor by which we multiply each term to get the next one. In the provided sequence:
  • -2
  • \( \frac{2}{3} \)
  • -\( \frac{2}{9} \)
To find the common ratio \( r \), you divide the second term by the first term: \[ r = \frac{\frac{2}{3}}{-2} = -\frac{1}{3} \] Always double-check your result by dividing the third term by the second term: \[ r = \frac{-\frac{2}{9}}{\frac{2}{3}} = -\frac{1}{3} \] Both checks confirm the common ratio is \( -\frac{1}{3} \).
general term
The general term, often denoted as \( a_n \), provides a formula to find any term in the sequence. For a geometric sequence, it's derived using the first term \( a \) and the common ratio \( r \). The general term formula is: \[ a_n = a \times r^{(n-1)} \]
Substituting the values from the given sequence, where \( a = -2 \) and \( r = -\frac{1}{3} \), we get: \[ a_n = -2 \times \left(-\frac{1}{3}\right)^{(n-1)} \] This formula means if you need, say the 4th term, plug in \( n = 4 \) and calculate.
geometric progression
A geometric progression is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In the context of our example:
  • Starting term: -2
  • Common ratio: \( -\frac{1}{3} \)
The progression is formed as follows:
If you start with -2 and keep multiplying by \( -\frac{1}{3} \), you'll get the terms: -2, \( \frac{2}{3} \), -\( \frac{2}{9} \) and so forth. Using the general term helps in efficiently finding any term in this progression without computing all previous terms. It streamlines calculations especially for large numbers.

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Most popular questions from this chapter

Find the values of a and \(d\) by solving each system. \(a+7 d=12\) \(a+2 d=7\)

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