Problem 213 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

If \(a, b, c\) be in A.P. \(b, c, a\) be in H.P. then prove that \(c, a, b\) are in G.P.

Step-by-Step Solution

Answer

If \(a, b, c\) are in A.P and \(b, c, a\) are in H.P, then \(c, a, b\) are indeed in G.P as proved.

Step 1: Understanding the meaning of A.P, H.P and G.P.

Firstly, we must understand the meanings of an arithmetic progression (A.P.), harmonic progression (H.P.), and geometric progression (G.P.). An A.P. is a sequence of numbers in which the difference of any two successive numbers is a constant. An H.P. is a sequence of numbers which are reciprocals of an arithmetic progression. A G.P. is a sequence 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.

Step 2: Finding the relations

Since a, b, c are in A.P, \(b - a = c - b\). Also, since \(b, c, a\) are in H.P, \(\frac{1}{c} - \frac{1}{b} = \frac{1}{b} - \frac{1}{a}\). This implies that \(2b = a + c\) and \(\frac{2}{b} = \frac{1}{a} + \frac{1}{c}\)

Step 3: Proving that \(c, a, b\) are in G.P.

We need to show that \(a = \sqrt{c \cdot b}\) to prove \(c, a, b\) are in G.P. Now, using the relationship derived in step 2, we get \(a = \sqrt{(2b - a) \cdot b}\). As we substitute \(a = 2b - a\) into \(\frac{2}{b} = \frac{1}{a} + \frac{1}{c}\), we get \(c = 2b - a\). Hence, \(a = \sqrt{(2b - a) \cdot b}\) is indeed true, therefore, \(c, a, b\) are indeed in G.P.

Key Concepts

Harmonic Progression (H.P.)

Harmonic Progression (H.P.) is an interesting sequence where each term after the first is derived by taking the reciprocal of an Arithmetic Progression (A.P.). To be clear, in an A.P., the difference between consecutive terms is constant. In an H.P., however, we are dealing with the reciprocals of these terms. This means that if you have an A.P. like:

• 2, 4, 6, 8...

Then an H.P. would be:

• 1/2, 1/4, 1/6, 1/8...

To show whether numbers are in an H.P., we ensure that the reciprocal values maintain the characteristics of an A.P. Mathematically, if three numbers \(b, c, a\) are in H.P., then the relationship between them can be expressed as: \[\frac{1}{c} - \frac{1}{b} = \frac{1}{b} - \frac{1}{a}\]. This rearranges to give a useful equation for problem-solving: \(\frac{2}{b} = \frac{1}{a} + \frac{1}{c}\). This useful characteristic helps us bridge conditions from sequences in H.P. to other types, like G.P.

Geometric Progression (G.P.)

A Geometric Progression (G.P.) is a captivating progression where each term after the first is calculated by multiplying the previous term by a constant non-zero value, known as the common ratio. For example, consider a G.P. expressed as:

• 3, 6, 12, 24...

In this example, every number is obtained by multiplying the previous one by 2, which is the common ratio.
We can mathematically represent a G.P. as \(ar, ar^2, ar^3,...\) where \(a\) is the first term and \(r\) is the common ratio. If you want to confirm that a sequence \(c, a, b\) is a G.P., you verify that \(a = \sqrt{c \cdot b}\). This means that

• \(a^2 = c \cdot b\)

Capturing the essence of multiplication-based sequences lets us transition naturally between other forms, such as A.P. or H.P., while always keeping an eye on the relationship between terms.

Mathematical Proofs

Mathematical proofs are structured arguments that validate the truth of a given statement. Proofs are essential in mathematics as they provide a logical foundation for why something is true. In our example exercise, we used given conditions from A.P. and H.P. to establish the truthfulness of a G.P. formed by rearranging terms.For instance, when you need to prove a specific sequence, you start with known equations. Here, from the condition of A.P., which gives us \(b - a = c - b\), and from the H.P., where \(\frac{2}{b} = \frac{1}{a} + \frac{1}{c}\), these relationships set the stage for drawing further conclusions.The strength of proofs lies in how they connect different mathematical properties. For example, the idea that \(2b = a + c\) aids us in understanding how an H.P. condition converts to a G.P., because it connects directly to the logic needed to demonstrate that \(a^2 = c \cdot b\), proving the arrangement of terms in a G.P. Hence, mathematical proofs are vital in confirming the interrelatedness of these progressions.