Question

Insert a geometric mean between 7 and \(112 .\)

Short answer

The geometric mean between 7 and 112 is 28.

Step-by-step solution

Understand the Concept of a Geometric Mean

The geometric mean between two numbers, a and b, is the number x such that a and x form a geometric progression with b. This means that x is the square root of the product of a and b, which can be written as x = √(a*b).

Calculate the Geometric Mean

In this case, the two numbers are 7 and 112. To find the geometric mean x between them, calculate x = √(7*112).

Simplify and Find the Geometric Mean

Simplify the expression to find x. So, x = √(784). Then, calculate the square root of 784 to get the geometric mean. The square root of 784 is 28. Therefore, the geometric mean between 7 and 112 is 28.

Key concepts

Geometric Progression

Among the different types of numerical sequences that exist, one of the most intriguing is the geometric progression. A geometric progression, or geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. This series creates an exponential pattern that increases or decreases rapidly depending on the common ratio's value.

For instance, if you start with the number 2 and use a common ratio of 3, the geometric progression would be 2, 6, 18, 54, and so forth, each time multiplying by 3. The relationship between consecutive terms is what defines this sequence. It's fascinating to see how quickly the numbers escalate, showcasing the power of exponential growth or decay in various real-world scenarios such as population studies, finance, and other scientific applications.

When you insert a geometric mean between two numbers, you essentially create a mini geometric progression where the inserted number maintains the common ratio between the original two numbers. This creates a harmonious set of three numbers in geometric progression.

Square Root

The square root is a mathematical operation that may seem daunting at first, but it is truly a fundamental concept that is quite simple to understand. The square root of a number is a value that, when multiplied by itself, gives the original number. Think of it like solving a mystery: if a number were disguised in a square, finding the square root is the key to unveiling its true identity.

For example, consider the number 25. What number can you multiply by itself to get 25? The answer is 5, because 5 multiplied by 5 is indeed 25. Therefore, the square root of 25 is 5, usually written as √25=5. In our particular exercise example, the square root of 784 was required to find the geometric mean, which turned out to be 28 since 28 times 28 equals 784. This operation is elemental in many areas of mathematics and beyond, including geometry, algebra, and statistics.

Arithmetic and Geometric Sequences

While geometric sequences multiply by a consistent ratio, arithmetic sequences take a different approach—instead of multiplying, you add a constant value known as the 'common difference'. An arithmetic sequence is as straightforward as it sounds: starting with an initial term, each succeeding term is obtained by adding the common difference to the previous term.

For instance, starting at 3 with a common difference of 4, you would get a sequence like this: 3, 7, 11, 15, and so on, each time adding 4. The simplicity of an arithmetic sequence is what makes it easy to grasp and powerful to use, especially because it is seen in everyday life, such as the even spacing of fence posts or evenly timed traffic lights.

Understanding the difference between arithmetic and geometric sequences is crucial as it allows you to recognize patterns, solve problems effectively, and apply this knowledge in real-world situations. By grasping the concepts of increasing at a steady rate or growing exponentially, students can better analyze and interpret the growth or decline in diverse scenarios.