Problem 27 - Algebra 2

Question

Find the missing term of each geometric sequence. It could be the geometric mean or its opposite. $$ 12, \square, 3, \dots $$

Step-by-Step Solution

Answer

The missing term is 6 or -6.

Step 1: Identify the known values

The known values from the sequence are the first term (12) and the third term (3).

Step 2: Apply the formula for geometric mean

The formula for the geometric mean is \(\sqrt{ab}\). Apply this formula by substituting 'a' with the first term (12) and 'b' with the third term (3).

Step 3: Simplify

Perform the operations inside the square root first, then take the square root. The solution to \( \sqrt{12 \times 3} \) is \( \sqrt{36} = 6\) as the positive solution. For the negative solution, we use -6 which is the opposite as stated in the question.

Key Concepts

Geometric Mean

Understanding the geometric mean is crucial when working with sequences, especially in problems like finding missing terms. The geometric mean of two numbers, say a and b, is the number located between them in a geometric progression. It is computed using the formula:

• \( \sqrt{a \times b} \)

The geometric mean is particularly useful in sequences where the ratio between consecutive terms is constant. In the example given, with terms 12 and 3, the geometric mean helps to ascertain the missing middle term. Calculating \( \sqrt{12 \times 3} \) gives us 6, which is the positive geometric mean for these terms. The negative counterpart, -6, is also a solution due to the direction indicated by the problem statement.

Square Root

The square root is an operation that helps determine the original base number, whose square (obtained by multiplying the number by itself) gives your given number. In the example exercise, to apply the geometric mean formula, one calculates the square root of the product of the given terms 12 and 3.

• Firstly, multiply the two terms: 12 times 3 equals 36.
• Then, obtain the square root: \( \sqrt{36} \).This mathematical operation involves finding a number which, when squared, results in 36.The square root of 36 is 6.

In geometric sequences, square roots are frequently employed to bridge the gap between known terms. This is particularly evident when determining missing middle terms, where the square root of the product provides a median term.

Sequence Terms

A sequence in mathematics is a set of numbers in a specific order. Each member is called a term. In a geometric sequence, each term is derived by multiplying the previous term by a fixed, non-zero number, named the common ratio. In the given exercise involving the terms 12, \( \square \), and 3, these numbers form the initial part of a geometric sequence.

• The first term is 12, and the third term provided is 3.
• Using the geometric mean formula, the missing second term can be deduced to maintain the proportionality of a geometric sequence.
• The solution was found via the formula \( \sqrt{12 \times 3} = 6 \), indicating that 6 could be the missing term, keeping a constant ratio across the terms.Additionally, -6 is the alternative solution as acknowledged by the exercise's wording to include the opposite solution.

Thus, understanding the place and calculation of sequence terms allows for completing sequences systematically and logically.