Question

Find the sum of 3 geometric means between \(\frac{1}{3}\) and \(\frac{1}{48}(\mathrm{r}>0)\). (1) \(\frac{1}{4}\) (2) \(\frac{5}{24}\) (3) \(\frac{7}{24}\) (4) \(\frac{1}{3}\)

Short answer

Answer: (3) \(\frac{7}{24}\).

Step-by-step solution

Find the common ratio

Since there are 3 geometric means between the first term \(a_1 = \frac{1}{3}\) and the last term \(a_5 = \frac{1}{48}\), we can find the common ratio (r) by using the formula \(a_n = a_1 \times r^{n-1}\). Let's rewrite the formula for \(a_5\): \(a_5 = a_1 \times r^{5-1}\). Now substitute the given values: \(\frac{1}{48} = \frac{1}{3} \times r^4\). Solve for r: \(r^4 = \frac{1}{16}\) \(r = \sqrt[4]{\frac{1}{16}} = \frac{1}{2}\) Now we have the common ratio, r = \(\frac{1}{2}\).

Find the three geometric means

Now that we know the common ratio, we can find the three geometric means in the sequence. The second term (\(a_2\): the first geometric mean) can be found using the formula: \(a_2 = a_1 \times r = \frac{1}{3} \times \frac{1}{2} = \frac{1}{6}\). Similarly, we find the third term (\(a_3\): the second geometric mean): \(a_3 = a_2 \times r = \frac{1}{6} \times \frac{1}{2} = \frac{1}{12}\). Finally, we find the fourth term (\(a_4\): the third geometric mean): \(a_4 = a_3 \times r = \frac{1}{12} \times \frac{1}{2} = \frac{1}{24}\).

Find the sum of the three geometric means

Now that we have all three geometric means (\(a_2 = \frac{1}{6}\), \(a_3 = \frac{1}{12}\), and \(a_4 = \frac{1}{24}\)), we can find their sum: Sum = \(a_2 + a_3 + a_4\) Sum = \(\frac{1}{6} + \frac{1}{12} + \frac{1}{24}\) To sum these fractions, we find a common denominator, which is 24: Sum = \(\frac{4}{24} + \frac{2}{24} + \frac{1}{24}\) Sum = \(\frac{7}{24}\) So, the sum of the three geometric means between \(\frac{1}{3}\) and \(\frac{1}{48}\) is \(\frac{7}{24}\). Therefore, the correct answer is (3) \(\frac{7}{24}\).

Key concepts

Geometric Progression

When we talk about a geometric progression (GP), we are referring to 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 type of sequence can be easily identified by its multiplicative pattern.

For example, consider the sequence 2, 4, 8, 16, and 32. This sequence is a GP because every term is twice its preceding term - in other words, the common ratio here is 2. Understanding GPs is crucial for various mathematical concepts, including compound interest in finance, population growth models in biology, and many more areas.

Coming back to our exercise, we had to find the three geometric means between \(\frac{1}{3}\) and \(\frac{1}{48}\), which essentially meant finding the terms of a specific GP that fits this setup. Recognizing the structure of a GP is the first step in solving many problems related to geometric sequences.

Common Ratio Calculation

The common ratio of a geometric progression is the 'multiplier' that takes us from one term to the next in the sequence. Calculating this ratio is a pivotal step in understanding the behavior of the GP as a whole.

In our exercise, to calculate the common ratio, we used the formula \(a_n = a_1 \times r^{n-1}\), where \(a_n\) is the nth term, \(a_1\) is the first term, and \(r\) is the common ratio. Using this formula, we established a relationship between the first term, the nth term (which is known in the problem), and the common ratio. Solving this equation provided us the value of \(r\), which we then used to find the intervening geometric means. This process showcases how the common ratio is at the heart of defining a GP's characteristics and enables us to extrapolate or interpolate values within the sequence.

Geometric Sequence Summation

Summing the terms of a geometric sequence involves adding each of the terms within the sequence together. When dealing with a finite number of terms, as in our exercise, this can be done simply by calculating each term and then adding them up.

To find the sum of multiple terms in a GP, we often rely on the sequence's properties and our ability to find a common denominator if the terms are fractions, as they were in this particular exercise. After calculating each geometric mean using our common ratio, we added the terms together, ensuring the denominators were the same to get the correct sum: \(\frac{7}{24}\).

If we were dealing with an infinite GP, a different formula would be used to find the sum, emphasizing the difference in approach depending on the nature of the sequence. Regardless of the sequence's length, understanding how to sum its terms is an essential skill for solving many practical problems in mathematics.