If the p^th , q^th and r^? terms of a G.P. are a, b and c, respectively. Prove that a^q-r b^r-

step 1 number 22 questions say the p quoth and rth term of a gpr a b and c and to prove that that okay

If the p^th , q^th and r^? terms of a G.P. are a, b and c,...

Key Concepts

Nth Term Expression

The expression for the nth term of a geometric progression is given by a_n = a? · r^(n-1), where a? is the first term and r is the common ratio. This formula is critical because it enables the representation of terms located at any position within the sequence, thereby facilitating the comparison and manipulation of these terms through their indices.

Properties of Exponents

Understanding the laws of exponents is crucial when working with expressions in a geometric progression. These properties include rules such as multiplying like bases by adding exponents, and the power of a power rule, which states that (x^a)^b = x^(a·b). These principles allow for the effective manipulation and simplification of exponential expressions that arise in problems involving GP.

Manipulation of Exponential Equations

When dealing with expressions that involve exponents in a geometric progression, it is often necessary to strategically manipulate the powers and indices to reveal underlying relationships. This process includes combining like terms, simplifying ratios, and ultimately using exponent rules to cancel out terms, thereby proving the desired equality or identity.

Geometric Progression (GP)

A geometric progression is a sequence where each term is obtained by multiplying the previous term by a fixed, non-zero constant known as the common ratio. This concept is foundational because it provides the framework to express any term of the sequence in terms of the first term and the common ratio, which is essential for relating terms at different positions.

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