Question

If \(\mathrm{m}, \mathrm{n}, \mathrm{s}, \mathrm{t}\) are in GP, then \(\frac{1}{\mathrm{~m}}, \frac{1}{\mathrm{n}}, \frac{1}{\mathrm{~s}}, \frac{1}{\mathrm{t}}\) are in (a) HP (b) AGP (c) AP (d) \(\mathrm{GP}\)

Short answer

Answer: The reciprocals \(\frac{1}{m}, \frac{1}{n}, \frac{1}{s}, \frac{1}{t}\) also form a Geometric Progression (GP).

Step-by-step solution

Identify the common ratio for the given GP

Since m, n, s, and t form a GP, there exists a common ratio r such that:\[ \begin{split} \frac{n}{m} = r, \, \frac{s}{n} = r, \, \frac{t}{s} = r \end{split} \]

Find the relationship between terms in the sequence of reciprocals

We will now find the relationships between terms in the sequence: \(\frac{1}{m}, \frac{1}{n}, \frac{1}{s}, \frac{1}{t}\).\[ \begin{split} \frac{\frac{1}{m}}{\frac{1}{n}} = \frac{n}{m}, \, \frac{\frac{1}{n}}{\frac{1}{s}} = \frac{s}{n}, \, \frac{\frac{1}{s}}{\frac{1}{t}} = \frac{t}{s} \end{split} \]

Comparing relationships

We can observe that:\[\frac{1}{m}:\frac{1}{n} = \frac{n}{m}, \, \frac{1}{n}:\frac{1}{s} = \frac{s}{n}, \, \frac{1}{s}:\frac{1}{t} = \frac{t}{s}\]These are the same relationships as in #Step 1#, where we found the common ratio r for m, n, s, and t. Thus, the reciprocals have the same common ratio as the original GP.

Conclusion

Since the reciprocals have the same common ratio, r, the progression for the reciprocals \(\frac{1}{m}, \frac{1}{n}, \frac{1}{s}, \frac{1}{t}\) is in a Geometric Progression (GP). Therefore, the correct answer is (d) GP.

Key concepts

Harmonic Progression

When dealing with sequences, a Harmonic Progression (HP) is a specific type that can initially seem tricky to understand. Essentially, a sequence is in HP if the reciprocals of its terms form an Arithmetic Progression (AP). To put it simply, let's consider a sequence like \(a, b, c\). It's in HP if \(\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\) is in AP. This means the differences between consecutive reciprocals are constant.
In terms of problem-solving, knowing that a sequence is in HP allows us to use the properties of an AP to find unknown terms in the HP. For example, if you're given two terms of a harmonic sequence, you could find the third one by finding the AP formed by their reciprocals. Keep in mind, HPs are less common in everyday applications but are important in physics and engineering when dealing with wave frequencies and rates.

Arithmetic Progression

An Arithmetic Progression (AP) is one of the most straightforward mathematical sequences. The key characteristic of an AP is a constant difference between consecutive terms. For instance, in the sequence \(2, 5, 8, 11\), each term increases by 3. This consistent change is known as the "common difference."
The formula for the \(n\)-th term of an AP can be expressed as: \(a_n = a_1 + (n-1)\cdot d\), where \(a_1\) is the first term and \(d\) is the common difference.
APs are everywhere in real life, from calculating repayments on a loan, to planning equal intervals in scheduling. Understanding APs is foundational because many complex mathematical concepts, such as geometric series and calculus, build upon this idea of consistent change.

Common Ratio

In a Geometric Progression (GP), the concept of a "common ratio" is central. A sequence is in GP if each term after the first is found by multiplying the previous one by a constant, known as the common ratio. For example, in the sequence \(3, 6, 12, 24\), each term is multiplied by 2, making 2 the common ratio.
Given some terms \(m, n, s, t\) in GP with a common ratio \(r\), the terms satisfy \(\frac{n}{m} = \frac{s}{n} = \frac{t}{s} = r\). This consistency defines the sequence's mathematical structure. Understanding the common ratio allows us to determine any term in the sequence using the formula: \(a_n = a_1 \cdot r^{n-1}\), where \(a_1\) is the first term and \(n\) is the term number.
Recognizing and calculating the common ratio is important in fields like finance, where it helps in calculating compound interest or analyzing exponential growth patterns.