Question

Three numbers form an increasing G.P. If the middle number is doubled, then the new numbers are in A.P. The common ratio of the G.P. is : (a) \(2-\sqrt{3}\) (b) \(2+\sqrt{3}\) (c) \(\sqrt{3}-2\) (d) \(3+\sqrt{2}\)

Short answer

Answer: (b) \(2+\sqrt{3}\)

Step-by-step solution

Create variables for G.P. and A.P. sequences

Let's assign variables to the numbers in the G.P. Let the first number be \(a\), the second number be \(ar\), and the third number be \(ar^2\), where \(r\) is the common ratio. For the A.P. series, let the new second number be \(2ar\). Now we have two sequences: G.P. - \(a, ar, ar^2\) and A.P. - \(a, 2ar, ar^2\).

Use the Arithmetic Progression property

Since the new numbers form an A.P, we can write down the condition for A.P.: \(2(2ar) = a + ar^2\).

Simplify the equation

Simplify the equation from step 2: \(4ar = a + ar^2\). Divide both sides of the equation by \(a\), and we get: \(4r = 1 + r^2\)

Rearrange the equation to find the common ratio \(r\)

Rearrange the equation from step 3: \(r^2 - 4r + 1 = 0\). Now solve this quadratic equation for the common ratio \(r\). Use the quadratic formula: \(r = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}\), where \(a = 1\), \(b = -4\), and \(c = 1\).

Solve for \(r\) using the quadratic formula

Plug in the values of \(a\), \(b\), and \(c\) to get the common ratio \(r\): \(r = \dfrac{4 \pm \sqrt{(-4)^2-4(1)(1)}}{2(1)}\).

Simplify the expression

Simplify it to get \(r = 2 \pm\sqrt{3}\). As the G.P. is increasing, the common ratio should be greater than 1. Thus, select the option \(r=2+\sqrt{3}\). The answer is (b) \(2+\sqrt{3}\).

Key concepts

Arithmetic Progression

An arithmetic progression (AP) is a sequence of numbers in which the difference between any two successive members is a constant. This constant is known as the common difference, denoted by 'd'. For instance, in the sequence 3, 5, 7, 9, ..., the common difference is 2, because each term after the first is 2 more than the previous term.

The general formula for the nth term of an AP is given by:
\( a_n = a_1 + (n-1)d \), where \( a_1 \) is the first term and \( n \) is the number of terms. In the context of the exercise, doubling the middle term of the G.P. to form an A.P. is a transformation that involves finding a common difference that satisfies the new sequence, effectively establishing a relation between the arithmetic and geometric progressions involved.

Common Ratio

The common ratio in a geometric progression (G.P.) is the factor by which each term is multiplied to obtain the next term in the sequence. For example, in the G.P. 2, 6, 18, 54, ..., each term is multiplied by 3 to get to the next term, making 3 the common ratio.

In a G.P., any term divided by its preceding term gives you the common ratio, mathematically represented as \( r = \frac{a_{n}}{a_{n-1}} \) where \( a_{n} \) is the nth term. During the exercise, we see that upon doubling the middle term of the G.P., the sequence transforms, necessitating a property check to ascertain if it then forms an A.P. This leads to the formation of a quadratic equation to solve for the common ratio.

Quadratic Equation

A quadratic equation is a second-degree polynomial equation in a single variable x, with a ≠ 0. It has the standard form \( ax^2 + bx + c = 0 \), where a, b, and c are constants. This equation has two solutions, which can be real or complex, and are given by the quadratic formula:
\( x = \frac{-b \rpm \tqrt{b^2-4ac}}{2a} \).

The discriminant, \( b^2-4ac \), determines the nature of the roots. If it is positive, there are two real and distinct roots. If it is zero, there is one real root (a repeated root). If it is negative, the roots are complex. In our exercise, we solve a quadratic equation to find the common ratio of the G.P., demonstrating the interconnectedness of these mathematical concepts.