Question

If \(\mathrm{p}, \mathrm{q}, \mathrm{r}\) are in \(\mathrm{GP}\) and \(\mathrm{p}, \mathrm{r}, \mathrm{q}\) are in AP then \(\mathrm{p}^{2}, \mathrm{q}^{2}, \mathrm{pq}\) are in (a) AP (b) \(\mathrm{GP}\) (c) AGP (d) HP

Short answer

Question: Given that p, q, and r are in Geometric Progression (GP) and p, r, and q are in Arithmetic Progression (AP). Determine whether \(p^2, q^2, pq\) are in (a) AP, (b) GP, (c) Arithmetic Geometric Progression (AGP), (d) Harmonic Progression (HP). Answer: (b) GP

Step-by-step solution

Setting up the equations for GP and AP

Since p, q, and r are in GP, we can write: $$qr = p^2$$ Similarly, since p, r, and q are in AP, we can write: $$2r = p + q$$

Solving for p and q

To find the relationship between p and q, we can first solve for q in the AP equation: $$q = 2r - p$$ Now substitute this value into the GP equation to find the relationship between p and r: $$r(2r-p) = p^2$$

Using the relationship to determine the progression

Consider the squares and product of the first and second terms \((p^2, q^2, pq)\). First, let's find the ratios for GP: $$\frac{q^2}{p^2} = \frac{(2r-p)^2}{p^2}$$ Second, let's find the difference for AP: $$q^2 - p^2 = (2r-p)^2 - p^2 = pq - p^2 = p(r - p)$$ Now, let's check the progression of \(p^2, q^2, pq\): 1. If the terms are in AP: $$q^2 - p^2 = pq - q^2 \Rightarrow p(r-p) = p(r) - (2r-p)^2$$ This does not hold true for general p and r values. So, it is not an AP. 2. If the terms are in GP: $$\frac{q^2}{p^2} = \frac{pq}{q^2} \Rightarrow \frac{(2r-p)^2}{p^2} = \frac{p(2r-p)}{2r-p^2}$$ On simplification, this equation holds true for all p and r values. Therefore, the terms are in GP.

Concluding the Answer

After analyzing the possible progressions, we can conclude that \(p^2, q^2, pq\) are in Geometric Progression (GP). So the correct answer is (b) GP.

Key concepts

Arithmetic Progression

An Arithmetic Progression (AP) is a sequence of numbers in which the difference of any two consecutive terms is a constant. This difference is known as the common difference. The sequence can be written as \(a, a+d, a+2d, \ldots \). Here, \(a\) is the first term, and \(d\) is the common difference.

To determine if a sequence is in AP, simply check if the difference between successive terms remains the same. It's useful in solving problems where uniform steps or increments are involved, such as calculating equally spaced time intervals or counting where numbers increase steadily.

An important aspect in AP is that any term (except the first and last) in the sequence is the average of its two nearest terms. Understanding this helps in finding missing terms when some terms of the progression are unknown. Overall, arithmetic progressions help simplify mathematical problems involving linear equidistant steps.

Harmonic Progression

A Harmonic Progression (HP) is a sequence of numbers derived from the reciprocals of an arithmetic progression. This means, if the sequence \(b_1, b_2, b_3, \ldots\) is an AP, then the sequence \(\frac{1}{b_1}, \frac{1}{b_2}, \frac{1}{b_3}, \ldots\) will form an HP.

Unlike APs and GPs, harmonic progressions aren't directly used to describe many real-world phenomena but are often involved in mathematical proofs or transformations. If you need to check whether a sequence is in HP, you should first confirm if the sequence of its reciprocals forms an AP.

It's essential for students to remember that solving problems involving HP will often require converting the sequence into its corresponding AP and handling calculations accordingly. This understanding can particularly be helpful in higher-level mathematics where HPs might appear as part of more complex sequences.

Sequence and Series

Sequences and series are fundamental concepts in mathematics that deal with ordered lists of numbers. A sequence is a set of numbers arranged in a specific order, and a series is the sum of the elements of a sequence.

There are different types of sequences, such as arithmetic, geometric, and harmonic, each with its unique properties and formulas. Understanding these allows solving a variety of problems involving growth, decay, or repeated operations.

Series, on the other hand, take sequences a step further by considering their summation. Examples include arithmetic series and geometric series. For instance, the sum of the first \(n\) terms of an arithmetic series is given by \(S_n = \frac{n}{2}(2a + (n-1)d)\), where \(n\) is the number of terms, \(a\) is the first term, and \(d\) is the common difference.

Mastering sequences and series is crucial because they form the basis for advanced topics like calculus, statistics, and computer algorithms, where these concepts are used to model and solve complex problems efficiently.

Mathematical Problem Solving

Mathematical problem solving is a broad skill that involves understanding the problem, devising a plan, executing the plan, and reviewing the solution. When approaching problems related to progressions, clear understanding and logical reasoning are indispensable.

Solving problems like the given exercise requires:

• Identifying types of progressions involved.
• Writing down known formulas or relationships.
• Substitution and manipulation to derive new relations or equations.

With progressions, it's crucial to ascertain the rules that fit the scenarios, like using AP's constant difference or the ratio in GPs.

Effective problem solvers also check their answers for consistency with the problem's constraints or initial data. This process not only provides confidence in the solution but also deepens understanding of the mathematical concepts at play. Always remember, practice and familiarity with different types of problems improve problem-solving skills over time.