Problem 35 - A Complete Resource Book in Mathematics for JEE Main

Question

Suppose \(a, b, c\) are in A.P. and \(a^{2}, b^{2}, c^{2}\) are in G.P. If \(a

Step-by-Step Solution

Answer

The value of \(a\) is \(\frac{1}{2\sqrt{2}}\) (option A).

Step 1: Understanding the Context

Given that \( a, b, c \) are in Arithmetic Progression (A.P.). This implies \( b - a = c - b = d \), where \(d\) is the common difference. Also, \(a^2, b^2, c^2\) are in Geometric Progression (G.P.), implying \( \left( \frac{b^2}{a^2} \right) = \left( \frac{c^2}{b^2} \right) \). Finally, \( a + b + c = \frac{3}{2} \) and \( a < b < c \).

Step 2: Express Terms using the Common Difference

Represent \(b\) and \(c\) in terms of \(a\) and \(d\) using the A.P. condition: \( b = a + d \) and \( c = a + 2d \). This gives:\[ a + (a + d) + (a + 2d) = \frac{3}{2} \]Simplifying, we find:\[ 3a + 3d = \frac{3}{2} \]Thus, \(a + d = \frac{1}{2}\).

Step 3: Use the G.P. Condition

Apply the G.P. condition on the squares: \( \left( \frac{b^2}{a^2} \right) = \left( \frac{c^2}{b^2} \right) \). Substitute \( b = a+d \) and \( c = a+2d \):\[ \left( \frac{(a + d)^2}{a^2} \right) = \left( \frac{(a + 2d)^2}{(a + d)^2} \right) \]Expand and simplify both sides to yield the equality:\[ \frac{a^2 + 2ad + d^2}{a^2} = \frac{a^2 + 4ad + 4d^2}{a^2 + 2ad + d^2} \].

Step 4: Solve for the Relationship

Upon cross-multiplying and simplifying, derive the equation:\[ (a^2 + 2ad + d^2)^2 = a^2(a^2 + 4ad + 4d^2) \]This simplifies down to a relationship involving \(a\) and \(d\). Substituting \( a + d = \frac{1}{2} \) into these terms helps narrow down possible values of \(a\) and \(d\).

Step 5: Determine the Value of 'a'

Use the fact that \(a < b < c\) and the numeric specific constraint \( a + b + c = \frac{3}{2} \). Given that \(a + d = \frac{1}{2} \), solve for \(a\) to maintain all conditions. After checking the values, it resolves to \( a = \frac{1}{2\sqrt{2}} \).

Key Concepts

Geometric Progression

In mathematics, a sequence of numbers is said to be in a Geometric Progression (G.P.) when each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. This means that if you have a sequence like \( a^2, b^2, c^2 \), to be in G.P., there must exist a constant ratio such that \( \frac{b^2}{a^2} = \frac{c^2}{b^2} \). When solving problems involving G.P., it's crucial to understand:

• The common ratio can be found by dividing any term by the previous term.
• Calculating the terms using formulas like \( a_n = a_1 \, r^{n-1} \), where \( a_1 \) is the first term and \( r \) is the common ratio.

In this exercise, the condition that \( a^2, b^2, c^2 \) are in a geometric progression is used to form an equation involving their squares, which helps us solve for the individual terms in the sequence.

Common Difference

The Common Difference is a crucial aspect of an Arithmetic Progression (A.P.). This is because in an A.P., the difference between consecutive terms is always the same and is called the common difference. If you have the sequence \( a, b, c \) in A.P., then find that \( b - a = c - b = d \), where \( d \) represents the common difference.

• The common difference is the key to expressing terms in relation to each other, like \( b = a + d \) and \( c = a + 2d \).
• Understanding the common difference allows the simplicity of solving equations involving arithmetic sequences as it simplifies the expression of terms to function only of the initial term \( a \) and the common difference \( d \).

In the given problem, recognizing the sequence \( a, b, c \) as an A.P. allows us to relate the terms and combine the provided conditions \( a + b + c = \frac{3}{2} \) effectively. This straightforward relationship becomes crucial when combined with the G.P. condition for \( a^2, b^2, c^2 \).

Algebraic Equations

Algebraic Equations involve variables and coefficients and are instrumental in forming relationships that describe mathematical concepts. When dealing with progressions, these equations help us to solve for unknowns like \( a \) and \( d \).

• In our problem, equations are set by substituting the terms from the progressions into algebraic formulas, such as \( b = a + d \) and \( c = a + 2d \).
• The equality of the G.P. condition \( \frac{b^2}{a^2} = \frac{c^2}{b^2} \) becomes one such equation, allowing further manipulation and simplification.
• Solving these involves expanding, simplifying, and substituting back any conditions like \( a + d = \frac{1}{2} \) until we isolate \( a \).

In quite complex problems, especially those involving both A.P. and G.P., grasping the algebraic manipulation is essential. This ensures you achieve the result efficiently, as demonstrated here, where algebraic equations connected the conditions helping to find \( a = \frac{1}{2\sqrt{2}} \).