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Chapter 11: Problem 11

Find the common ratio in each geometric sequence. $$10^{2}, 10^{3}, 10^{4}, \dots$$

Short Answer

Expert verified
The common ratio is 10.

Step by step solution

01

- Identify Terms in the Sequence

Identify the first few terms of the geometric sequence. Here they are: \[ 10^{2}, 10^{3}, 10^{4}, \text{ and so on} \]
02

- Select Two Consecutive Terms

Choose any two consecutive terms in the sequence. Let's take the first two terms: \[ a_{1} = 10^{2} \] and \[ a_{2} = 10^{3} \]
03

- Define the Common Ratio

The common ratio \( r \) is found by dividing the second term by the first term. \[ r = \frac{a_{2}}{a_{1}} \]
04

- Calculate the Common Ratio

Substitute the values of the first two terms into the formula: \[ r = \frac{10^{3}}{10^{2}} = 10 \]

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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 factor that you multiply by, to get from one term to the next. Specifically, it is the constant number you use to find succeeding terms using the current term. For example, if the common ratio is 2, you multiply the current term by 2 to get the next term. The significance of the common ratio lies in its ability to consistently progress the sequence. It can be a positive number, a negative number, or even a fraction, thereby adjusting the growth or decay of the sequence. Without knowing the common ratio, you cannot predict further terms in the geometric sequence.
terms in geometric sequence
A geometric sequence is a list 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. For instance, in the sequence \[ 10^2, 10^3, 10^4 \] the terms are 100, 1000, and 10000 respectively. Each term follows a specific order, and each subsequent term is generated by applying the common ratio to the previous term. The first term is typically denoted as \( a_1 \), the second term as \( a_2 \), and so forth. Understanding geometric sequences helps in decoding exponential growth patterns seen in various real-life contexts like population growth, radioactive decay, and investment returns.
calculating common ratio
Calculating the common ratio in a geometric sequence is straightforward. Here is how you do it:
  • Choose any two consecutive terms in the sequence.
  • For illustration, let's consider the sequence \[ 10^2, 10^3, 10^4 \]. Choose the first two terms: \( a_1 = 10^2 \) and \( a_2 = 10^3 \).
  • Divide the second term by the first term to get the common ratio: \[ r = \frac{a_2}{a_1} = \frac{10^3}{10^2} = 10 \]
By following these steps, you've found that the common ratio is 10. This ratio indicates how much each term is multiplied by to obtain the next term. Repeating this for any two consecutive terms will consistently yield the same common ratio, reinforcing the inherent structure of the geometric sequence.

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