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Chapter 14: Problem 8

If the given sequence is geometric, find the common ratio \(r\). If the sequence is not geometric, say so. See Example 1 . $$ \frac{2}{3},-\frac{2}{15}, \frac{2}{75},-\frac{2}{375}, \dots $$

Short Answer

Expert verified
The sequence is geometric with common ratio \(r = -\frac{1}{5}\).

Step by step solution

01

Identify the Sequence

First, identify the given terms of the sequence: \(\frac{2}{3}\), \(-\frac{2}{15}\), \(\frac{2}{75}\), \(-\frac{2}{375}\), ...
02

Calculate the Ratios

Calculate the ratio between consecutive terms to check if the sequence is geometric. Compute the ratio \(r\) for the first pair of terms: \(r = \frac{-\frac{2}{15}}{\frac{2}{3}} = -\frac{2}{15} \times \frac{3}{2} = -\frac{1}{5}\).
03

Verify the Ratios

Compute the ratios for the subsequent pairs of terms to ensure they are the same. For the second and third terms: \(r = \frac{\frac{2}{75}}{-\frac{2}{15}} = \frac{2}{75} \times -\frac{15}{2} = -\frac{1}{5}\).
04

Additional Verification

Verify the ratio for the third and fourth terms: \(r = \frac{-\frac{2}{375}}{\frac{2}{75}} = -\frac{2}{375} \times \frac{75}{2} = -\frac{75}{375} = -\frac{1}{5}\).
05

Conclusion

Since the ratio \(r\) is consistently \(-\frac{1}{5}\) for all consecutive pairs, the sequence is geometric with common ratio \(r = -\frac{1}{5}\).

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Key Concepts

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

Common Ratio
The common ratio is a fundamental concept in geometric sequences. It represents the factor by which each term in the sequence is multiplied to get the next term. In a geometric sequence, the common ratio (denoted as \( r \)) remains constant for all consecutive terms. For example, consider the sequence provided: \( \frac{2}{3}, -\frac{2}{15}, \frac{2}{75}, -\frac{2}{375}, \dots \). To find the common ratio, divide each term by its preceding term.

For the first pair of terms, the ratio \( r \) is calculated as:
\( r = \frac{-\frac{2}{15}}{\frac{2}{3}} = -\frac{1}{5} \). This calculation is repeated for all consecutive terms to ensure they yield the same common ratio.
Geometric Progression
A geometric progression (or geometric sequence) is a sequence where each term is obtained by multiplying the previous term by a fixed, non-zero number called the common ratio. In our sequence, starting from \( \frac{2}{3} \), each subsequent term is formed by multiplying by \( -\frac{1}{5} \).

Let's break it down further:
  • First term: \( \frac{2}{3} \)
  • Second term: \( \frac{2}{3} \times -\frac{1}{5} = -\frac{2}{15} \)
  • Third term: \( -\frac{2}{15} \times -\frac{1}{5} = \frac{2}{75} \)
  • And so on...

The sequence continues by repeatedly applying the common ratio.
Term Calculation
To calculate any term in a geometric sequence, we use the formula:
\[ a_n = a_1 \times r^{n-1} \]
where \( a_n \) is the \( n \)th term, \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the term number.

For our sequence:
  • First term \( a_1 = \frac{2}{3} \)
  • Common ratio \( r = -\frac{1}{5} \)
To find the fourth term:
\[ a_4 = \frac{2}{3} \times \left(-\frac{1}{5}\right)^{4-1} = \frac{2}{3} \times \left(-\frac{1}{5}\right)^3 = \frac{2}{3} \times -\frac{1}{125} = -\frac{2}{375} \]
Term calculation using this formula helps to find any term in the sequence without needing to compute all previous terms.
Sequence Verification
Verification helps ensure a sequence is geometric by confirming the consistency of the common ratio across all pairs of terms. This is done by:
  • Calculating the common ratio \( r \) between each pair of consecutive terms.
  • Ensuring the calculated \( r \) is the same for all pairs in the sequence.
This step is essential because, for a sequence to be geometric, the ratio needs to remain constant.

In our example, we calculated:

  • Between the first and second terms: \( r = -\frac{1}{5} \)
  • Between the second and third terms: \( r = -\frac{1}{5} \)
  • Between the third and fourth terms: \( r = -\frac{1}{5} \)
Since the ratio \( r = -\frac{1}{5} \) is consistently the same across all pairs, it confirms that our sequence is indeed geometric.

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