Question

If the second number of three numbers in an increasing \(\mathrm{GP}\) is doubled, we get an \(\mathrm{AP}\), then (a) common ratio of the GP is 1 or \(-1\) (b) common ratio of the GP and common difference of AP are equal. (c) common difference of AP is equal to both first term and last term. (d) common ratio of the GP is \(2+\sqrt{3}\).

Short answer

(a) Common ratio of the GP is 1 or -1. (b) Common ratio of the GP and common difference of AP are equal. (c) Common difference of AP is equal to both first term and last term. (d) Common ratio of the GP is 2+√3. Answer: (a) False (b) False (c) False (d) True

Step-by-step solution

Define the terms of the GP

Let the terms of the geometric progression be \(a, ar\), and \(ar^2\), where \(a\) is the first term, \(r\) is the common ratio, and the progression is increasing.

Apply the AP condition

According to the given condition, if we double the second term, we get an arithmetic progression. Therefore, we can write: \(a, 2ar, ar^2\). The common difference for this arithmetic progression will be \(2ar - a\).

Determine the common difference of the AP

Since it is an arithmetic progression, the difference between the third and second term should be equal to the difference between the second and first term. So, we have: \(ar^2 - 2ar = 2ar - a\). Now, solve this equation to find \(r\).

Solve for r

Move all terms involving r to one side: \(ar^2 - 4ar + a = 0\) Factor out \(a\): \(a(r^2 - 4r + 1) = 0\) Since \(a\) cannot be 0 as it is the first term of an increasing GP, we can safely divide both sides by \(a\): \(r^2 - 4r + 1 = 0\) Recognize that \((r-2)^2 = r^2 - 4r + 4\). Therefore, our equation can be rewritten as: \((r-2)^2 - 3 = 0\) Now, solve for \(r\): \((r-2)^2 = 3\) \(r-2 = \pm \sqrt{3}\) \(r = 2 \pm \sqrt{3}\) So, the common ratio of the GP is \(2+\sqrt{3}\) and \(2-\sqrt{3}\).

Evaluate the given statements

(a) common ratio of the GP is 1 or \(-1\) False, because the common ratio is \(2+\sqrt{3}\) and \(2-\sqrt{3}\). (b) common ratio of the GP and common difference of AP are equal. False, because the common ratio of the GP is \(2+\sqrt{3}\) or \(2-\sqrt{3}\), but the common difference in the AP is \(2ar - a\). (c) common difference of AP is equal to both first term and last term. False, because the common difference of the AP is \(2ar - a\), which is not equal to the first term \(a\), or the last term \(ar^2\). (d) common ratio of the GP is \(2+\sqrt{3}\). True, because one of the possible values for the common ratio is indeed \(2+\sqrt{3}\). Therefore, the correct answer is only option (d).

Key concepts

Arithmetic Progression

An arithmetic progression (AP) is a sequence of numbers in which the difference between consecutive terms is constant. This difference is known as the common difference.
For example, in the sequence 3, 6, 9, 12, ..., the common difference is 3. If you subtract any term from the one that follows it, you'll get this constant value of 3.
This pattern makes arithmetic progressions easy to work with, as each term can be expressed by adding the common difference to the previous term.

• The nth term of an AP can be found using the formula: \(a_n = a + (n - 1) imes d\), where \(a\) is the first term and \(d\) is the common difference.
• APs are linear in nature because the change is steady and predictable.

Common Ratio

In a geometric progression (GP), the common ratio is the factor by which we multiply each term to get to the next term in the sequence.
It determines how quickly the sequence grows or shrinks.
If the common ratio is greater than 1, the sequence grows; if it's between 0 and 1, the sequence shrinks. A negative common ratio results in the alternating signs of sequence terms.

• For example, in the GP 2, 4, 8, 16, ..., the common ratio is 2.
• We can calculate the common ratio \(r\) using the formula: \(r = \frac{a_{n+1}}{a_n}\), where \(a_n\) is any term in the sequence and \(a_{n+1}\) is the following term.

In the problem, the common ratio is critical for understanding how the sequence is altered to become an arithmetic progression when the second term is doubled.

Common Difference

The common difference is a key feature of arithmetic progressions. It's the fixed amount added to each term of the sequence to find the next term.
For APs, it stays constant, making formulas for calculating sequences much simpler.
In an AP, each term is derived from the previous one by consistently adding this common difference.

• Given the AP: 5, 10, 15, 20, ..., the common difference \(d\) is 5.
• This is determined by subtracting a term from the next: \(d = 10 - 5 = 5\).

In the provided exercise, understanding the common difference is vital to establishing a connection between modified terms of the geometric progression and converting them into an arithmetic progression.

Quadratic Equation

Quadratic equations are polynomial equations of the form \(ax^2 + bx + c = 0\). They are primarily used to find two possible values (roots) for variables that satisfy the equation.
Quadratics can describe a wide range of phenomena in mathematics and physics.

• The standard formula for solving a quadratic equation is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
• These solutions reveal the points where a parabolic graph intersects the x-axis.

In this exercise, the concept of a quadratic equation arises when solving for the common ratio.
The terms are rearranged into the format suitable to solve as a quadratic equation, leading to two potential solutions for the common ratio of the GP.