Problem 9 Find the common ratio for each g... [FREE SOLUTION]

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Chapter 14: Problem 9

Find the common ratio for each geometric sequence. $$ 10,20,40,80, \dots $$

Short Answer

Expert verified

The common ratio is 2.

Step by step solution

Identify the First Term

The first term of the geometric sequence is 10.

Identify the Second Term

The second term of the geometric sequence is 20.

Calculate the Common Ratio

To find the common ratio, divide the second term by the first term. \[\text{Common Ratio} = \frac{20}{10} = 2\]

Verify the Ratio

To ensure correctness, confirm that dividing any term by the previous term yields the same common ratio. For example, \[\frac{40}{20} = 2 \text{ and } \frac{80}{40} = 2\]

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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, one of the key concepts is the 'common ratio'. This is the factor by which each term in the sequence increases or decreases. To calculate the common ratio, you divide any term by the term immediately before it. For example, in the sequence: 10, 20, 40, 80,..., you take the second term (20) and divide it by the first term (10). This gives you a common ratio of $\frac{20}{10} = 2$. This ratio must be consistent across the sequence for it to be classified as geometric.

First Term

Determining the first term of a geometric sequence is the initial step, as it serves as the starting point for generating all subsequent terms. In our example sequence: 10, 20, 40, 80,...., the first term is 10. This first term is fundamental because each future term is derived by repeatedly multiplying it by the common ratio. Always identify the first term before proceeding to calculate or verify other terms.

Sequence Verification

Verification is crucial to ensure accuracy in identifying a geometric sequence. Once you have calculated the common ratio, it is important to verify that each term divided by the previous term yields exactly the same common ratio. For our example sequence: 10, 20, 40, 80,..., we initially found the common ratio to be 2. To verify, we can perform checks: $\frac{40}{20} = 2$ and $\frac{80}{40} = 2$. Consistent results confirm that our sequence is indeed geometric with a common ratio of 2.

Division

Division is a fundamental arithmetic operation used to calculate the common ratio in a geometric sequence. When you have a sequence, the common ratio is found by dividing one term by the term immediately preceding it. For instance, taking 20 (second term) and dividing it by 10 (first term) in our sample sequence (10, 20, 40, 80,...), the division gives us 2. Mastery of division is essential as it underlies the method for confirming the uniformity of a geometric sequence.

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91影视

Find the common ratio for each geometric sequence. $$ 10,20,40,80, \dots $$

Short Answer

Step by step solution

Identify the First Term

Identify the Second Term

Calculate the Common Ratio

Verify the Ratio

Key Concepts

Common Ratio

First Term

Sequence Verification

Division

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