If a, b, c are in G.P. and loga-log2 b, log2 b-log3 c and...

If $a, b, c$ are in G.P. and $\log a-\log 2 b, \log 2 b-\log 3 c$
and $\log 3 c-\log a$ are in A.P., then $a, b, c$ are the sides of a triangle which is
(A) right angled
(B) acute angled
(C) obtuse angled
(D) None of these

Key Concepts

Logarithmic Properties

Logarithmic properties, such as the identity that the difference of logarithms is equal to the logarithm of the quotient, are fundamental tools for transforming multiplicative relationships into additive ones. This conversion simplifies the handling of sequences involving ratios and differences, and it is key in solving equations where logarithms are used to relate different variables.

Triangle Classification and Properties

Understanding the properties of triangles, including the triangle inequality and the criteria for classifying triangles as acute, right, or obtuse, is crucial in geometry. These classifications depend on the relationships among the side lengths and angles of the triangle, and are essential for determining the nature of a triangle given certain algebraic conditions on its sides.

Geometric Progression

A geometric progression is a sequence of numbers where each term after the first is found by multiplying the previous term by a common constant, known as the common ratio. This concept is essential for understanding relationships among quantities that change multiplicatively, which is a key idea in many areas of mathematics including solving equations that rely on proportional relationships.

Arithmetic Progression

An arithmetic progression is a sequence of numbers in which the difference between consecutive terms is constant. This concept applies to situations where quantities increase or decrease by a fixed amount over equal intervals, and it is crucial for setting up conditions in problems that involve evenly spaced differences or balancing symmetric relationships.

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