(a) a, b, c are positive real numbers different from 1. If...

(a) $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are positive real numbers different from 1. If $\log _{2}{ }^{100}, 2 \log _{\mathrm{b}}{ }^{10}, 2 \log _{\mathrm{c}}{ }^{5}+\log _{\mathrm{c}}{ }^{4}$ (p) AP are in $\mathrm{HP}$, then $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in (b) $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are different positive real numbers such that $\mathrm{a}>\mathrm{b}>\mathrm{c}$. If $2 \log (\mathrm{a}-\mathrm{c}), \log$ (q) GP $\left(a^{2}-c^{2}\right), \log \left(a^{2}+2 b^{2}+c^{2}\right)$ are in $A P$, then $a, b, c$ are in (c) If $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in $\mathrm{AP}$ and $\mathrm{a}^{2}, \mathrm{~b}^{2}, \mathrm{c}^{2}$ are in HP then, $\mathrm{a}^{3}, \mathrm{~b}^{3}, \mathrm{c}^{3}$, are in (r) HP (d) If $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in $\mathrm{AP}, \mathrm{b}, c, \mathrm{~d}$ are in $\mathrm{GP}, \mathrm{c}, \mathrm{d}, \mathrm{e}$ are in $\mathrm{HP}$ then $\mathrm{a}, \mathrm{c}, \mathrm{e}$ are in (s) AGP

(a) $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are positive real numbers different from 1. If $\log _{2}{ }^{100}, 2 \log _{\mathrm{b}}{ }^{10}, 2 \log _{\mathrm{c}}{ }^{5}+\log _{\mathrm{c}}{ }^{4}$
(p) AP are in $\mathrm{HP}$, then $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in
(b) $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are different positive real numbers such that $\mathrm{a}>\mathrm{b}>\mathrm{c}$. If $2 \log (\mathrm{a}-\mathrm{c}), \log$
(q) GP $\left(a^{2}-c^{2}\right), \log \left(a^{2}+2 b^{2}+c^{2}\right)$ are in $A P$, then $a, b, c$ are in
(c) If $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in $\mathrm{AP}$ and $\mathrm{a}^{2}, \mathrm{~b}^{2}, \mathrm{c}^{2}$ are in HP then, $\mathrm{a}^{3}, \mathrm{~b}^{3}, \mathrm{c}^{3}$, are in
(r) HP
(d) If $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are in $\mathrm{AP}, \mathrm{b}, c, \mathrm{~d}$ are in $\mathrm{GP}, \mathrm{c}, \mathrm{d}, \mathrm{e}$ are in $\mathrm{HP}$ then $\mathrm{a}, \mathrm{c}, \mathrm{e}$ are in
(s) AGP

Key Concepts

Arithmetic-Geometric Progression (AGP)

An arithmetic-geometric progression is a sequence in which each term is the product of the corresponding terms of an arithmetic progression and a geometric progression. This hybrid sequence captures both additive and multiplicative progression characteristics, making it a complex yet interesting structure that combines elements from both APs and GPs. It finds applications in various advanced mathematical problems, particularly those involving mixed growth patterns.

Sequence Transformation Relations

Sequence transformation relations involve establishing links between different types of progressions—such as between arithmetic, geometric, and harmonic progressions—by applying algebraic operations or functional transformations (e.g., taking reciprocals or logarithms). This approach provides a deeper understanding of the intrinsic relationships between sequences and allows one to deduce properties of one sequence type from another.

Logarithm Properties

Logarithms are the inverses of exponential functions and they have several useful properties, such as the product, quotient, and power rules. These properties allow complex multiplicative and exponential relationships to be transformed into simpler additive or linear forms. In problems where logarithms are applied to sequence terms, these properties often facilitate the conversion of multiplicative relations into additive ones that may align with arithmetic progressions.

Harmonic Progression (HP)

A harmonic progression is a sequence of numbers whose reciprocals form an arithmetic progression. This transformation between the terms of an HP and an AP is particularly useful, as it allows problems involving harmonic sequences to be addressed by applying familiar techniques from arithmetic progressions. The concept is important in various fields including statistics and physics.

Geometric Progression (GP)

A geometric progression is a sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. This multiplicative relationship allows the use of exponentiation in expressing the nth term and summing the series, and it plays a crucial role in many algebraic and number theory problems.

Arithmetic Progression (AP)

An arithmetic progression is a sequence of numbers in which the difference between successive terms is constant. This constant difference, known as the common difference, makes it straightforward to predict subsequent terms and to derive expressions for the nth term or the sum of a finite number of terms. The concept is foundational in algebra and is widely used in problems involving evenly spaced values.

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