Column I Column II (a) If a^x=b^γ=c^x where a, b, c are in GP, then x, y, z are in (p) AP (b)

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Column I Column II (a) If a^x=b^γ=c^x where a, b, c are in...

Column I $\quad$ Column II (a) If $\mathrm{a}^{x}=\mathrm{b}^{\gamma}=\mathrm{c}^{x}$ where $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in $\mathrm{GP}$, then $\mathrm{x}, y, \mathrm{z}$ are in (p) AP (b) Three distinct numbers $a, b, c$ satisfying $\frac{1}{b-a}+\frac{1}{b-c}=\frac{1}{a}+\frac{1}{c}$ are in (q) $\mathrm{GP}$ (c) If $\mathrm{x}, \mathrm{y}, \mathrm{z}(\mathrm{all}>1)$ are in GP, then $\frac{1}{1+\log \mathrm{x}}, \frac{1}{1+\log \mathrm{y}}, \frac{1}{1+\log \mathrm{z}}$ are in (r) HP (d) Three consecutive terms $\frac{1}{1+\sqrt{x}}, \frac{1}{1-x}, \frac{1}{1-\sqrt{x}}$ of a sequence are in (s) AGP

Column I $\quad$ Column II
(a) If $\mathrm{a}^{x}=\mathrm{b}^{\gamma}=\mathrm{c}^{x}$ where $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in $\mathrm{GP}$, then $\mathrm{x}, y, \mathrm{z}$ are in
(p) AP
(b) Three distinct numbers $a, b, c$ satisfying $\frac{1}{b-a}+\frac{1}{b-c}=\frac{1}{a}+\frac{1}{c}$ are in
(q) $\mathrm{GP}$
(c) If $\mathrm{x}, \mathrm{y}, \mathrm{z}(\mathrm{all}>1)$ are in GP, then $\frac{1}{1+\log \mathrm{x}}, \frac{1}{1+\log \mathrm{y}}, \frac{1}{1+\log \mathrm{z}}$ are in
(r) HP
(d) Three consecutive terms $\frac{1}{1+\sqrt{x}}, \frac{1}{1-x}, \frac{1}{1-\sqrt{x}}$ of a sequence are in
(s) AGP

Key Concepts

Exponential and Logarithmic Functions in Sequence Transformations

Exponential functions and their inverses, logarithms, are key tools in converting multiplicative relationships (such as those found in geometric progressions) into additive ones (as seen in arithmetic progression). This transformation is fundamental in solving and analyzing equations where terms are expressed in exponential form.

Geometric Progression (GP)

A geometric progression is a sequence in which each term after the first is obtained by multiplying the previous term by a fixed, non-zero constant called the common ratio. Understanding GP is crucial, especially when dealing with exponential relationships and transformations into other forms of sequences.

Arithmetic Progression (AP)

An arithmetic progression is a sequence where the difference between consecutive terms is constant. This concept is important in algebraic manipulations where linear relationships are established, and it often appears when logarithms are used to transform multiplicative relations into additive ones.

Harmonic Progression (HP)

A harmonic progression is defined as a sequence of numbers whose reciprocals form an arithmetic progression. This transformation highlights the interplay between different types of sequences and is particularly useful when working with reciprocal expressions or solving equations that involve fractions.

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