Problem 130 - A Complete Resource Book in Mathematics for JEE Main

Question

\begin{tabular}{l} Column-I & Column-II \\ \hline I. If \(a, b, c\) are in A.P., \(b, c, d\) are in G.P. and \(c, d, e\) are in H.P., then (A) A.P. \\ \(\qquad a, c, e\) are in \\ II. If \(2(y-a)\) is the H.M. between \(y-x, y-z\) then \(x-a, y-a, z-a\) & (B) G.P. \\\ are in \\ III. If three numbers are in H.P., then the numbers obtained by subtract- (C) H.P. \\ ing half of the middle number from each of them are in \\ IV. If \(a, b, c\) are in G.P., then the equations \(a x^{2}+2 b x+c=0\) and \(d x^{2}+\) (D) A.G.P. \\ \(2 e x+f=0\) have a common root, if \(\frac{d}{a}, \frac{e}{b}\) and \(\frac{f}{c}\) are in \end{tabular}

Step-by-Step Solution

Answer

I - (A), II - (A), III - (A), IV - (D)

Step 1: Solve Statement I

Given that \(a, b, c\) are in Arithmetic Progression (A.P.), \(b, c, d\) are in Geometric Progression (G.P.), and \(c, d, e\) are in Harmonic Progression (H.P.). We need to determine the relationship for \(a, c, e\). In A.P., we have the condition \(2b = a + c\). In G.P., the condition is \(c^2 = bd\). In H.P., we have \(\frac{2de}{d+e} = c\). By substituting and solving these equations, we get that \(a, c, e\) form an A.P. Therefore, Statement I matches with (A) A.P.

Step 2: Solve Statement II

Given that \(2(y-a)\) is the Harmonic Mean (H.M.) between \(y-x\) and \(y-z\). This implies the formula for H.M.: \[2(y-a) = \frac{2(y-x)(y-z)}{(y-x)+(y-z)}\]Solving for \(x-a, y-a, z-a\), the expression simplifies to an Arithmetic Progression (A.P.) since then the conditions \((x-a) + (z-a) = 2(y-a)\) must hold true. Therefore, Statement II matches with (A) A.P.

Step 3: Solve Statement III

For three numbers in Harmonic Progression (H.P.): \(a, b, c\), and subtracting half of the middle number \(\frac{b}{2}\), results in \(a-\frac{b}{2}, 0, c-\frac{b}{2}\). This transformation yields an Arithmetic Progression (A.P.) since the condition \(b = \frac{a+c}{2}\) holds because of the subtraction shifting terms equally. Thus, Statement III matches with (A) A.P.

Step 4: Solve Statement IV

Given \(a, b, c\) are in Geometric Progression (G.P.) and the equations have a common root if \(\frac{d}{a}, \frac{e}{b}, \frac{f}{c}\) are in an Arithmetic-Geometric Progression (A.G.P.), which is a sequence of the form \(a, ar+d, ar^2+2d\). This corresponds with the condition that links the equations through the common root property, showing that (D) A.G.P. is correct for Statement IV.

Key Concepts

Arithmetic Progression (A.P.)

An arithmetic progression (A.P.) is a sequence of numbers in which the difference between consecutive terms remains constant. This difference is known as the 'common difference'. For example, in the sequence 2, 4, 6, 8, the common difference is 2.

In the context of the exercise, if numbers \( a, b, c \) are in A.P., it means that they satisfy the condition \( 2b = a+c \). This equation signifies that \( b \) is the average of \( a \) and \( c \).

• Any three terms \( a, b, c \) in A.P. satisfy: \( b-a = c-b \).
• This uniform spacing makes predicting subsequent terms straightforward.
• Finding the n-th term of an A.P. follows the formula: \( a_n = a_1 + (n-1)d \).
• The sum of the first n terms is given by: \( S_n = \frac{n}{2} (2a + (n-1)d) \).

Geometric Progression (G.P.)

A geometric progression (G.P.) is a sequence where each term after the first is the product of the previous term and a fixed, non-zero number known as the common ratio. For instance, in the sequence 3, 9, 27, 81, the common ratio is 3.

In the problem at hand, if \( b, c, d \) are in G.P., it means they adhere to the rule \( c^2 = bd \). This indicates that \( c \) is the geometric mean of \( b \) and \( d \).

• Many formulas relate G.P.s to other mathematical concepts such as exponential growth.
• The general form of a G.P. is: \( a, ar, ar^2, ar^3, \ldots \).
• Finding the n-th term is straightforward: \( a_n = ar^{n-1} \).
• The sum of the first n terms is given by: \( S_n = a \left( \frac{r^n - 1}{r - 1} \right) \) for \( r eq 1 \).

Harmonic Progression (H.P.)

A harmonic progression (H.P.) is a sequence of numbers derived from the reciprocals of an arithmetic sequence. This means if a sequence \( a, b, c \) is in H.P., then \( \frac{1}{a}, \frac{1}{b}, \frac{1}{c} \) forms an A.P.

The implications of this are neatly illustrated in the exercise where \( c, d, e \) are in H.P., satisfying the condition \( \frac{2de}{d+e} = c \). This equation shows that \( c \) is the harmonic mean of \( d \) and \( e \).

• Converting an H.P. to an A.P. is done through reciprocals: \( a, b, c \rightarrow \frac{1}{a}, \frac{1}{b}, \frac{1}{c} \).
• Useful for problems involving rates or proportions.
• Unlike others, the notion of "common difference" involves reciprocal values.
• Calculating the sum or the n-th term in H.P. is less straightforward compared to A.P. and G.P.