Problem 24 - Algebra 2

Question

Find the missing term of each geometric sequence. It could be the geometric mean or its opposite. $$ \frac{2}{5}, \square, \frac{8}{45}, \dots $$

Step-by-Step Solution

Answer

The missing term in the geometric sequence is \( \frac{4}{9} \).

Step 1: Understand the sequence

A geometric sequence is a sequence of numbers where each term present after the initial term is found by multiplying the preceding term by a fixed, non-zero number called the ratio. So, we need to find the ratio and then multiply the initial term with the ratio to know the missing term.

Step 2: Determine the ratio

From the question, let's use \( r \) to represent the ratio between terms, and the \( \square \) to represent the missing term. We can write \( r \frac{2}{5} = \square \) and \( r \square = \frac{8}{45} \). That's actually a system of 2 equations with 2 variables, which we can solve.

Step 3: Substitute and solve

Substitute \( r = \square / \frac{2}{5} \) into the second equation, we get \( \square^2 / \frac{2}{5} = \frac{8}{45} \). Then solve for \( \square \), we get \( \square = \sqrt{\frac{8}{45} * \frac{2}{5}} = \frac{4}{9}\)

Key Concepts

Geometric Mean

In mathematics, the geometric mean is a special type of average. It is used primarily in sequences known as geometric sequences. Imagine you have a sequence where you need to find a missing number between two known values. This missing number we find using the geometric mean.

• The geometric mean is found by multiplying the two numbers (before and after the missing term) and taking the square root of the result.
• For example, if you know two numbers are \( a \) and \( b \), and there is one missing term between them, then the geometric mean is \( \sqrt{a \times b} \).

This concept is particularly useful when dealing with geometric sequences, where each term is obtained by multiplying the previous term by a common ratio. Knowing how to find the geometric mean can help you solve problems involving missing terms in such sequences.

Ratio in Sequences

In a geometric sequence, the ratio is a key element. It is the constant factor by which each term is multiplied to arrive at the next term.

• For example, if you have a sequence \( \frac{2}{5}, \, \square, \, \frac{8}{45} \), the ratio needs to be found or used to find the missing middle term.
• Knowing the first term and the ratio allows you to systematically determine any term in the sequence.

To discover this ratio in a sequence, you divide a term by its preceding term. In cases where a term is missing, you can use equations that relate the known terms and the unknown term using the ratio. Understanding this helps in identifying and predicting future terms accurately in a sequence.

Solving Equations in Sequences

Sometimes, finding a missing term in a sequence involves solving equations. This requires setting up relations and then solving them.

• In our exercise, we derived the equations \( r \frac{2}{5} = \square \) and \( r \square = \frac{8}{45} \), where \( r \) is the ratio and \( \square \) is the unknown term.
• By substituting one equation into the other, we isolated \( \square \) in terms of known values and ultimately due to simplification, found \( \square = \frac{4}{9} \).

Understanding how to set up and solve these types of equations is crucial when working with sequences. It allows you to fill in the gaps and ensures you can continue a sequence correctly. Being comfortable with manipulating algebraic expressions to solve such systems is an invaluable skill in math.