If the p t h, q th and r th terms of both an A.P. and a G.P....

If the $p t h, q$ th and $r$ th terms of both an A.P. and a G.P. be respectively $a, b$ and $c$, then (A) $a^{e} \cdot c^{b} \cdot b^{a}=a^{c} \cdot b^{e} \cdot a^{b}$ (B) $a^{b-1} \cdot b^{e+1} \cdot c^{a-1}=a^{e-1} \cdot b^{a-1} \cdot c^{b+1}$ (C) $a^{b} \cdot b^{c} \cdot c^{a}=a^{c} \cdot b^{a} \cdot c^{b}$ (D) None of these

If the $p t h, q$ th and $r$ th terms of both an A.P. and a G.P. be respectively $a, b$ and $c$, then
(A) $a^{e} \cdot c^{b} \cdot b^{a}=a^{c} \cdot b^{e} \cdot a^{b}$
(B) $a^{b-1} \cdot b^{e+1} \cdot c^{a-1}=a^{e-1} \cdot b^{a-1} \cdot c^{b+1}$
(C) $a^{b} \cdot b^{c} \cdot c^{a}=a^{c} \cdot b^{a} \cdot c^{b}$
(D) None of these

Key Concepts

Arithmetic Progression

An arithmetic progression is a sequence in which each term after the first is obtained by adding a constant difference to the previous term. The nth term is given by a formula of the form first term plus (n-1) times the common difference. This concept is central in problems involving linear sequences and is key to understanding relations among terms positioned at different indices.

Geometric Progression

A geometric progression is a sequence where each term after the first is found by multiplying the previous term by a constant called the common ratio. The nth term is expressed as the first term multiplied by the common ratio raised to the (n-1)th power. This concept is fundamental in understanding exponential growth or decay in sequences and is often used when comparing with arithmetic progressions.

Nth Term Formulas

The nth term formulas for arithmetic and geometric progressions provide a systematic method to determine any term in the sequence, based on its position. In an arithmetic progression, the formula is structured as the first term plus (n-1) times the common difference, while in a geometric progression it is the first term multiplied by the common ratio raised to the (n-1) power. These formulas are crucial when equating or comparing terms from different types of sequences.

Relationship Between Sequences as A.P. and G.P.

When the same set of terms belong simultaneously to both an arithmetic progression and a geometric progression, it imposes special constraints on the values and indices of these terms. Such problems often require establishing a relationship between the common difference and the common ratio, yielding identities that link the sequence terms and their positions. This concept is important in solving and simplifying expressions that arise from the concurrent conditions.

Algebra of Exponents

Algebra of exponents involves the rules and properties governing exponential expressions, such as the power of a product, quotient, and power rules. Mastery of these properties is essential when dealing with expressions where terms appear as exponents, which is a common occurrence in problems that combine sequences with exponential functions. This understanding is pivotal in simplifying and comparing complex exponential expressions that feature in many advanced algebra problems.

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