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Chapter 11: Problem 12

Find the common ratio in each geometric sequence. $$10^{-1}, 10^{-2}, 10^{-3}, \ldots$$

Short Answer

Expert verified
The common ratio is \(10^{-1}\).

Step by step solution

01

Identify the First Term (\

The first term of the sequence is given as \(a_1 = 10^{-1}\). This is the starting point of the geometric sequence.
02

Identify the Second Term (\

The second term of the sequence is \(a_2 = 10^{-2}\). This is the term that follows the first term in the geometric sequence.
03

Determine the Common Ratio (\

The common ratio \(r\) can be found by dividing the second term by the first term: \(r = \frac{a_2}{a_1} = \frac{10^{-2}}{10^{-1}} = 10^{-2 - (-1)} = 10^{-2 + 1} = 10^{-1}\). Thus, the common ratio is \(10^{-1}\).

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Key Concepts

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

Common Ratio
In a geometric sequence, the common ratio is the number that each term is multiplied by to get the next term. Here, it鈥檚 found by dividing the second term by the first term. In our sequence, we see that the first two terms are \text{subterms of 10, namely} (10^{-1} and 10^{-2}). This leads us to compute the common ratio as follows: \( r = \frac{a_2}{a_1} = \frac{10^{-2}}{10^{-1}} = 10^{-2 - (-1)} \). After simplifying, we get \( r = 10^{-1} \). This value tells us how the sequence progresses, essentially showing the multiplication factor between terms. Typically, understanding and identifying the common ratio is key for mastering geometric sequences.
First Term
The first term of a geometric sequence is the initial value from which the sequence starts. In this problem, the first term is given as \( a_1 = 10^{-1} \). Identifying this first term is crucial because it sets the stage for the rest of the sequence. It helps to place the entire sequence in context and is used to determine both subsequent terms and principles such as the common ratio. Knowing the first term is like having the starting point on a map; once identified, it becomes much easier to navigate the entire sequence.
Second Term
The second term in a geometric sequence is the next term that follows the first term. Here, the second term is given as \( a_2 = 10^{-2} \). The second term is important because it helps in defining the common ratio. To figure out how the sequence progresses, you rely on this term along with the first term. Computing the common ratio involves taking this \( a_2 \) and dividing it by the first term \( a_1 \). This provides the factor by which each term multiplies to yield the next term, establishing the pattern of the sequence.
Mathematical Notation
Understanding mathematical notation is essential for mastering sequences like these. In a geometric sequence, a common notation for terms might be \( a_1, a_2, a_3, \, \dots \). The common ratio is often denoted by \( r \), while each term itself might be represented as \( a_n \) where \( n \) signifies the term number. The formulae used in sequences, such as \( r = \frac{a_{n+1}}{a_n} \), help us concise represent complex relationships clearly. It鈥檚 important to be comfortable with these notations, as they simplify the process of calculating and solving for terms within sequences. Proper use of notation leads to better understanding and more efficient problem-solving.

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