Question

If the 3rd, 7 th and 11 th terms of a geometric progression are \(\mathrm{p}, \mathrm{q}\) and \(\mathrm{r}\) respectively, then the relation among \(\mathrm{p}, \mathrm{q}\) and \(\mathrm{r}\) is (1) \(\mathrm{p}^{2}=\mathrm{qr}\) (2) \(\mathrm{r}^{2}=\mathrm{qp}\) (3) \(\mathrm{q}^{2}=\mathrm{p}^{2} \mathrm{r}^{2}\) (4) \(\mathrm{q}^{2}=\mathrm{pr}\)

Short answer

Question: Find the correct relationship between the 3rd, 7th, and 11th terms of a geometric progression, given as p, q, and r, respectively. Answer: The correct relationship between p, q, and r is \(q^2 = pr\).

Step-by-step solution

Solve for "a" and "r" in terms of p and q

We have the following equations: \(1) ar^2 = p\) \(2) ar^6 = q\) We need to express a and r in terms of p and q. We can start by solving for 'a' in equation 1. \(a = \frac{p}{r^2}\)

Plug the expression for "a" into the second equation

With "a" in terms of p and r, we can plug it back into the second equation to get the relationship between p, q, and r. \(\frac{p}{r^2}r^6 = q\) Simplify the equation: \(pr^4 = q\)

Substitute a back to the third term, which is related to "r" and simplify

Now, we use the expression for "a" to find the relationship with the 11th Term, which is r, given by \(3) ar^{10} = r\) Plug in the expression for "a": \(\frac{p}{r^2} r^{10} = r\) Simplify the equation: \(p r^8 = r\)

Rewrite the relationship between p, q, and r

We now have a relationship between p, r, and q: \(1) pr^4 = q\) \(2) p r^8 = r\) We can rewrite the first equation in terms of r by dividing both sides by p: \(r^4 = \frac{q}{p}\) Now we can multiply both the sides of this equation by r^4 to eliminate the fraction: \(r^8 = qr^4\) Now we can see that the relationship between p, q, and r is as follows: \(r^{8} = qr^{4}\), or similarly \(q = \frac{r^8}{r^4}\) However, we are missing how 'r' relates to 'p'. So, we can divide the second equation by the first equation to solve for r^4: \(\frac{pr^8}{pr^4} = \frac{r^4}{1}\) \(r^4 = \frac{q}{p}\)

Express the relationship in terms of p, q, and r

We now plug this expression for \(r^4\) back into the second equation: \(p\left(\frac{q}{p}\right) = qr\)

Simplify the final expression

Simplify the equation: \(q^2 = pr\) So the relation among p, q, and r is: \(\boxed{q^2 = pr}\), which corresponds to option (4).

Key concepts

Geometric Progression Terms

A geometric progression (GP), also known as a geometric sequence, is a series 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 (r). Understanding the terms in a geometric progression is key for solving many algebraic problems involving sequences.

For example, if we have the 3rd (p), 7th (q), and 11th (r) terms of a GP, then these terms are related by the powers of the common ratio, r. The nth term of a GP is given by the general form \(ar^{n-1}\), where 'a' is the first term of the progression. In the context of the exercise, knowing these relationships helps to express p, q, and r algebraically using powers of r to discover the underlying relationship among them.

Algebraic Equations

Algebraic equations are mathematical statements that assert the equality of two expressions. They consist of variables, coefficients, and constants. Having a good grip on manipulating these equations is crucial when dealing with geometric progression problems. The step-by-step solution provided makes use of several algebraic equations to solve for the variables 'a' (the first term) and 'r' (the common ratio).

By expressing the given terms of the sequence p, q, and r in the form of algebraic equations involving 'a' and 'r', and then solving for these variables, the students can decipher the complex relationship between different terms of the sequence. This also demonstrates the power of substitution and simplification in handling algebraic expressions to reveal insights that are not immediately apparent.

Geometric Sequence Properties

Geometric sequences exhibit specific characteristics that make them unique and predictable, which in turn allows for problem-solving strategies such as those used in the given exercise. Key properties of a geometric sequence include the constant ratio between consecutive terms, and the ability for any term to be calculated using the formula for the nth term: \(T_n = a r^{(n-1)}\).

In the exercise provided, understanding that any term in the sequence can be found by multiplying the first term by the common ratio raised to the power that is one less than the term's position leads to a system of equations that, when solved, reveal the necessary relationship. This is exemplified in the solution steps where the equations for p, q, and r are manipulated to find the sought-after relation: \(q^2 = pr\). Moreover, these properties are essential for translating practical scenarios into mathematical models that can be easily navigated using algebraic techniques.