If the sides a, b, c of ABC are in GP where loga-log2 b, log2 b-log3 c, log3 c-loga are in AP,

step 1 In this problem, A, B, C are in Gp. So I can write the value of B square is equal to A .C. Solvi

If the sides a, b, c of ABC are in GP where loga-log2 b,...

If the sides $\mathrm{a}, \mathrm{b}, \mathrm{c}$ of $\triangle \mathrm{ABC}$ are in GP where $\log \mathrm{a}-\log 2 \mathrm{~b}, \log 2 \mathrm{~b}-\log 3 \mathrm{c}, \log 3 \mathrm{c}-\log \mathrm{a}$ are in $\mathrm{AP}$, then the sides $\mathrm{a}$, b, c are in the ratio (a) $4: 6: 9$ (b) $9: 4: 6$ (c) $9: 6: 4$ (d) $3: 6: 8$

If the sides $\mathrm{a}, \mathrm{b}, \mathrm{c}$ of $\triangle \mathrm{ABC}$ are in GP where $\log \mathrm{a}-\log 2 \mathrm{~b}, \log 2 \mathrm{~b}-\log 3 \mathrm{c}, \log 3 \mathrm{c}-\log \mathrm{a}$ are in $\mathrm{AP}$, then the sides $\mathrm{a}$,
b, c are in the ratio
(a) $4: 6: 9$
(b) $9: 4: 6$
(c) $9: 6: 4$
(d) $3: 6: 8$

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

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 constant non-zero factor called the common ratio. In the context of triangle sides, if the sides are in a geometric progression, this means that the ratio between consecutive sides is constant, which provides a proportional relationship that can be used to derive further relations among the sides.

Arithmetic Progression

An arithmetic progression is a sequence in which the difference between consecutive terms is constant. When differences of logarithms are in arithmetic progression, it signifies that the logs of certain products or ratios maintain a consistent difference, which, upon exponentiation, leads to multiplicative relations among the original quantities.

Logarithmic Properties

Logarithmic properties, such as the fact that the difference of logarithms is the logarithm of a quotient, are used to transform additive relationships among logarithmic values into multiplicative relationships among the original numbers. This is especially useful when dealing with sequences in logarithmic form, as it allows for the conversion between arithmetic progression in the log-domain and multiplicative (or geometric) relationships in the original values.

Transcript

Please give Ace some feedback