If a, b, c are positive numbers in G.P. and log ((5 c)/(a)), log((3 b)/(5 c)) and log((a)/(3 b))

step 1 Hello everyone here we are given that b square is equal to AC and 2 log 3b by 5c is equal to log

If a, b, c are positive numbers in G.P. and log ((5 c)/(a)),...

If $a, b, c$ are positive numbers in G.P. and log $\left(\frac{5 c}{a}\right), \log \left(\frac{3 b}{5 c}\right)$ and $\log \left(\frac{a}{3 b}\right)$ are in A.P. then $a, b, c$ (A) form the sides of an equilateral triangle (B) form the sides of an isosceles triangle (C) form the sides of a right angled triangle (D) can not form the sides of a triangle

If $a, b, c$ are positive numbers in G.P. and log $\left(\frac{5 c}{a}\right), \log \left(\frac{3 b}{5 c}\right)$ and $\log \left(\frac{a}{3 b}\right)$ are in A.P. then $a, b, c$
(A) form the sides of an equilateral triangle
(B) form the sides of an isosceles triangle
(C) form the sides of a right angled triangle
(D) can not form the sides of a triangle

Key Concepts

Triangle Inequality Condition

The triangle inequality condition is a fundamental property in geometry which states that for three lengths to form a triangle, the sum of any two sides must exceed the third side. This is crucial when determining whether given numerical relationships among potential side lengths can indeed constitute the sides of a valid triangle.

Logarithm Properties

Logarithm properties are essential when transforming multiplicative relationships into additive ones. Key properties include the product rule (log(xy) = log x + log y), the quotient rule (log(x/y) = log x - log y), and the power rule (log(x^n) = n log x). These properties enable the conversion of ratios and products into expressions that are easier to handle in both arithmetic and geometric progression contexts.

Geometric Progression

A geometric progression (G.P.) is a sequence in which each term after the first is obtained by multiplying the preceding term by a constant called the common ratio. In problems involving G.P., the relation between terms can be expressed by equations such as b² = ac for three consecutive terms, establishing proportional relationships that are often leveraged to find unknown values or establish further properties among the terms.

Arithmetic Progression

An arithmetic progression (A.P.) is a sequence where the difference between successive terms is constant. When expressions or functions of variables (such as logarithms) are in A.P., it implies that twice the middle term equals the sum of the first and third terms. This property is used to set up equations that connect the different parts of the problem, facilitating the transition from logarithmic relations to algebraic equations.

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