Problem 38 - Algebra 2

Question

Find the missing terms of each geometric sequence. (Hint: The geometric mean of the first and fifth terms is the third term. Some terms might be negative.) 2.5, \(\quad\),\(\quad\),\(\quad\),\(202.5, \ldots\)

Step-by-Step Solution

Answer

The missing terms of the series are 7.5, 22.5, and 67.5

Step 1: calculate the geometric mean of first and fifth term

The geometric mean of the first and fifth terms would be equal to the third term in the series(i.e., the square root of the product of first term and fifth term). Let's denote the first term as 'a' and third term as 'c'. Hence, we can write this as \(c = \sqrt{a \times b}\). Substituting the given values: \(c = \sqrt{2.5 \times 202.5}\)

Step 2: calculate the common ratio

The common ratio of a geometric sequence is obtained by dividing any term by the preceding term. Since we already found the third term (geometric mean, 'c') and the first term is given (2.5), we calculate the common ratio (r) as \(r = c / a = \sqrt{2.5 \times 202.5} / 2.5\).

Step 3: find the missing terms

In a geometric progression, any term is the product of the preceding term and the common ratio. Therefore, the second term will be \(a \times r\), the third term will be \(a \times r^2\) , and the fourth term will be \(a \times r^3\). Hence, by substituting 'a' and 'r' into these equations and computing, we can obtain the missing terms in the series.

Key Concepts

Geometric Mean

The geometric mean is a useful concept when dealing with sequences where consecutive terms are multiplied by a constant to obtain the next term. In geometric sequences, the geometric mean of two numbers is particularly important. It is calculated as the square root of the product of those numbers. For example, if you have numbers 'a' and 'b', the geometric mean would be \( \sqrt{a \times b} \).
This concept is very practical when finding specific terms in geometric sequences. In the given exercise, the geometric mean of the first term (2.5) and the fifth term (202.5) gives the third term of the sequence. This property is a characteristic feature in geometric progressions, which allows us to find missing numbers efficiently.

Common Ratio

In a geometric sequence, the magic lies in the common ratio. This is the constant number you multiply by to get from one term to the next.
The common ratio is essential to determining all the terms in a sequence if you know a few of them. To find the common ratio, simply divide any term in the sequence by its preceding term. For instance, in our exercise, after finding the third term, the common ratio \( r \) is calculated as \( c / a \), where 'c' is the third term and 'a' is the first one. This ratio helps in propagating values across the sequence and ensures they adhere to the mathematical progression rules of a geometric series.

Missing Terms

To find the missing terms in a geometric sequence, we rely heavily on the first term and the common ratio. Knowing these, you can fill in any gaps in the series.
The process involves multiplying the known term by the common ratio to find the subsequent term. For example, if 'a' is the first term, and 'r' is the common ratio, then:

• The second term is \( a \times r \)
• The third term is \( a \times r^2 \)
• The fourth term is \( a \times r^3 \)

By following this pattern, you can systematically find all the missing numbers within the sequence, ensuring that the progression remains consistent and accurate.

Mathematical Progression

A mathematical progression is a sequence of numbers with a specific pattern. In a geometric progression, each term is generated by multiplying the previous one by a fixed number, known as the common ratio.
Geometric sequences are common in many real-world scenarios, including compound interest calculations and population growth models. They are distinguished from arithmetic progressions, where numbers are added by a constant.
Understanding both the foundational term and the common ratio is crucial in predicting the future terms in geometric sequences reliably. Such knowledge allows us to extend the sequence forward or backward without missing any important elements, making this concept fundamental in the study of sequences and series.