Problem 63 - Algebra 2

Question

Find the common ratio in the geometric sequence \(4,10,25,62.5, \ldots\) $$ \begin{array}{lllll}{\text { A. } 0.4} & {\text { B. } 2.5} & {\text { C. } 15} & {\text { D. } 25}\end{array} $$

Step-by-Step Solution

Answer

The common ratio in the geometric sequence is 2.5.

Step 1: Find the ratio between consecutive terms

The common ratio can be found by dividing the 2nd term in the sequence by the 1st term. So, the ratio is \( \frac{10}{4} = 2.5 \).

Step 2: Check correctness of the ratio

To ensure the ratio is correct, divide the 3rd term by the 2nd term. This gives \( \frac{25}{10} = 2.5 \). The result is same as before, verifying the ratio.

Step 3: Final check

To be thorough, divide the 4th term by 3rd term, \( \frac{62.5}{25} = 2.5 \). Again, we get 2.5 as the result, which confirms that the ratio is consistent across the sequence.

Key Concepts

Understanding Common Ratio

In a geometric sequence, the common ratio is a key factor that defines the sequence. It is a constant value that each term in the sequence is multiplied by to obtain the subsequent term. To find the common ratio, you divide one term by the preceding term. For our sequence:

• Divide the second term by the first term: \( \frac{10}{4} = 2.5 \)

This tells us the common ratio is 2.5. It's essential because it provides the framework on which all the terms in the sequence are based. The common ratio isn't just a number; it describes the pattern of growth along the sequence.
If the ratio is greater than 1, the sequence grows larger with each term. If it's between 0 and 1, the sequence decreases, and so forth.

Comprehending Consecutive Terms

Consecutive terms in a geometric sequence are sequential numbers that appear one after the other in the specified order. For example, in the sequence 4, 10, 25, and 62.5, these numbers are consecutive terms. Understanding their relationship is crucial:

• Each term is obtained by multiplying the previous term by the common ratio.
• Using our sequence, multiply: \( 4 \times 2.5 = 10 \), then \( 10 \times 2.5 = 25 \), and so on.

This consistent multiplication by the same ratio is what characterizes the uniform nature of a geometric sequence. By mastering the concept of consecutive terms, you can predict future numbers or verify past numbers in the sequence.

Sequence Verification Procedure

Sequence verification is a crucial step to ensure the calculated common ratio is accurate and applied consistently across the sequence. It's not enough to calculate the ratio once; you should double-check it between all consecutive terms. This process involves confirming that each term divided by its predecessor results in the same common ratio. Let's verify:

• From the second and third terms: \( \frac{25}{10} = 2.5 \)
• From the third and fourth terms: \( \frac{62.5}{25} = 2.5 \)

This verification process ensures consistency and accuracy throughout the sequence, affirming that each pair of consecutive terms aligns with the determined common ratio. Only when every consecutive pair checks out can you be confident that your geometric sequence is correctly established.