Problem 22 - Algebra 2

Question

Find the missing term of each geometric sequence. It could be the geometric mean or its opposite. $$ 5, \square, 911.25, \dots $$

Step-by-Step Solution

Answer

The missing term in the geometric progression is calculated to be: \(5 * r\) (where `r` is the common ratio as calculated in Step 1)

Step 1: Identify the Common Ratio

In a geometric sequence, the ratio of every pair of consecutive terms should stay consistent. This ratio is calculated by dividing a term by its previous term. The sequence provided does have a missing term, but since we are given the terms before and after the missing term, the common ratio 'r' could still be determined. Hence, calculate the square root of the ratio of the third term over the first term to find `r`. So, the common ratio is calculated as: \[ r = \sqrt{\frac{911.25}{5}} \]

Step 2: Find the Missing Term

Once we know the common ratio, identifying the missing term should be straight-forward. We can multiply the first term by the common ratio to find the second term (missing term in our case), as the characteristic of geometric sequence is that each term after the first is found by multiplying the previous term by a fixed, non-zero number. Hence, \[ \text{Missing term} = 5 * r \]

Step 3: Checking the Solution

In order to make sure our solution is correct, check if the missing number, when inserted into the sequence, maintains the constant ratio between all terms. This could be done by dividing the third term with the second term we just calculated and ensure if the result is same as our previously calculated common ratio `r`. If it is, then our solution is correct.

Key Concepts

Common Ratio

To understand geometric sequences, we must grasp the concept of the "common ratio." This is a fundamental property that defines how each term in the sequence relates to the previous one. In a geometric sequence, every term after the first is obtained by multiplying the preceding term by the common ratio.

• If we have the terms 5 and 911.25 in a sequence where the middle term is missing, we can still find the common ratio by considering these given terms.
• The common ratio ( ") is derived by taking the square root of the division of the known third term by the first term, in cases with a missing second term.

Using the example from the exercise, the formula is: \[ r = \sqrt{\frac{911.25}{5}} \] This calculation gives us the means to determine the middle, missing term by inversely calculating from the two outer terms.

Geometric Mean

The geometric mean in a sequence serves as an essential link between two terms in a geometric progression. When a term is missing in a geometric sequence, it's the geometric mean that may need to be calculated.

• The geometric mean is specifically relevant when solving for missing middle terms in a sequence of three terms.
• It acts as the bridge between the first and last term and maintains the consistent pattern affirmed by the common ratio.

In our given exercise example, the geometric mean allows us to discover the missing term. You calculate it essentially using the found common ratio by multiplying it with the known term (so in this case, multiply the first term 5 by the calculated common ratio). This step ensures that the sequence maintains its harmonious order.

Sequence Pattern

The characteristic sequence pattern of a geometric sequence determines how the terms follow one after another. Understanding this pattern involves recognizing the constant factor or multiplier, which is the common ratio.

• Every geometric sequence, by its nature, should maintain a constant ratio throughout its terms.
• The pattern is predictable; knowing one term and the common ratio can forecast the entire sequence.

In the exercise, the sequence pattern was slightly hidden due to the missing term. However, by applying the knowledge of geometric means and the common ratio, we fill in this gap seamlessly. Checking this pattern involves ensuring each term adheres to the calculated common ratio. Thus, once we find the missing term, validating it by checking the pattern continuity confirms our calculations and understanding of geometric progressions.