Problem 114 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(a, b, c\) are non-zero real numbers such that 3 \(\left(a^{2}+b^{2}+c^{2}+1\right)=2(a+b+c+a b+b c+c a)\), then, \(a, b, c\) are in (A) A.P. (B) G. P. (C) H.P. (D) all equal

Step-by-Step Solution

Answer

(A) A.P. and (D) all equal are true.

Step 1: Understanding the Problem

We are given a condition involving the squares and linear terms of the real numbers \(a\), \(b\), and \(c\). Our task is to determine which type of sequence these numbers form: Arithmetic Progression (A.P.), Geometric Progression (G.P.), Harmonic Progression (H.P.), or if all numbers are equal.

Step 2: Write Down the Given Equation

Start by writing the equation:3\(a^2 + b^2 + c^2 + 1 = 2(a + b + c + ab + bc + ca)\).

Step 3: Simplify the Equation

Distribute the terms to simplify the equation:3\(a^2 + b^2 + c^2 + 1 = 2a + 2b + 2c + 2ab + 2bc + 2ca\).

Step 4: Rearrange the Terms

Collect all terms on one side of the equation:3\(a^2 + b^2 + c^2 + 1\) - 2\(a + b + c + ab + bc + ca\) = 0.

Step 5: Testing Each Sequence Condition

Let's check for each sequence: -**A.P.:** \(b-a = c-b\) leads to \(a, b, c\) needing the form of equal differences.-**G.P.:** \(\frac{b}{a} = \frac{c}{b}\) leads to equal quotients.-**H.P.:** the reciprocals, \(\frac{1}{a}, \frac{1}{b}, \frac{1}{c}\), should form an A.P.-**All equal:** means \(a = b = c\).

Step 6: Substitute Conditions for A.P.

When \(a, b, c\) are in A.P., \(b = a+d\), \(c = a+2d\). Substitute these in the original equation and see if it holds true.

Step 7: Check with A.P. Substitution

Substitute \(b = a+d, c = a+2d\):- Left side: \(3(a^2 + (a+d)^2 + (a+2d)^2 + 1)\)- Right side: \(2(a + (a+d) + (a+2d) + a(a+d) + (a+d)(a+2d) + (a+2d)a)\)Both sides are calculated, and small values of \(a = 1, d = 0\) fit the equation.

Step 8: Verify Solutions for All Equal Case

Substituting \(a = b = c\) into the original equation also satisfies the equation perfectly, providing a valid scenario.

Key Concepts

Arithmetic Progression

An Arithmetic Progression (A.P.) is a sequence of numbers where the difference between consecutive terms remains constant. This difference is known as the 'common difference'. Imagine the numbers on a ruler: they go up evenly, like 2, 4, 6, 8.

• The formula for the n-th term of an A.P. is given by: \( a_n = a + (n-1) \cdot d \) where \( a \) is the first term and \( d \) is the common difference.
• To check if numbers are in A.P., simply verify if the difference \( b - a = c - b \) for any three terms \( a, b, c \).

In the given problem, if \( a, b, c \) satisfy the equation with a common difference, then they form an Arithmetic Progression. The step-by-step solution demonstrates checking this condition by substituting possible A.P. terms into the equation.

Geometric Progression

A Geometric Progression (G.P.) consists of numbers where each term after the first is found by multiplying the previous one by a fixed non-zero number called the 'common ratio'. Think of multiplying each number by 2: like 3, 6, 12, 24, etc.

• The formula for the n-th term of a G.P. is \( a_n = ar^{(n-1)} \), where \( a \) is the first term and \( r \) is the common ratio.
• For three numbers \( a, b, c \) to be in G.P., the relationship \( \frac{b}{a} = \frac{c}{b} \) must hold true.

The problem checks if \( a, b, c \) in G.P. satisfy the equation by looking for this consistent ratio in the terms given. If found, they form a Geometric Progression; however, the original solution shows alternative forms that fit better with A.P. or equality.

Harmonic Progression

A Harmonic Progression (H.P.) is closely related to an Arithmetic Progression, but instead of the numbers themselves, it's their reciprocals that form an A.P. For example, if 1, 1/2, 1/3, 1/4 form an A.P., 1, 2, 3, 4 forms an H.P.

• The sequences are determined by checking if \( \frac{1}{b-a} = \frac{1}{c-b} \), which translates to the reciprocals forming an A.P.
• This means the original sequence can be transformed into an A.P by taking reciprocals.

In the exercise, examining whether \( a^{-1}, b^{-1}, c^{-1} \) in A.P. could solve the equation would indicate if \( a, b, c \) form a Harmonic Progression. Yet, the answer shows checking through H.P. led to A.P. or equal results, simplifying the problem.

Equality of Numbers

The condition where all numbers are equal means each number in the sequence is exactly the same. This is the simplest form of progression where no differences or ratios are required since inequality doesn't exist.

• If \( a = b = c \), then every term remains the same, evidently simplifying the problem to basic arithmetic checks.
• This condition satisfies most equations easily as the symmetry makes balance effortless.

In the given context, the solution shows that substituting \( a = b = c \) in the equation checks out perfectly, meaning the scenario where all numbers are equal holds valid. Thus, it suggests that this might be the best fit if more complex sequence relations don't align with the original problem.