If a, b, c are three distinct numbers in AP and b-a, c-b, a are in GP, then a: b: c= (a) 2: 3: 1

step 1 Hello, so the question is taken from the sequence and series of your book. So given the followin

If a, b, c are three distinct numbers in AP and b-a, c-b, a...

If $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are three distinct numbers in $\mathrm{AP}$ and $\mathrm{b}-\mathrm{a}, \mathrm{c}-\mathrm{b}$, a are in $\mathrm{GP}$, then a: b: $\mathrm{c}=$
(a) $2: 3: 1$
(b) $\frac{1}{2}: \frac{1}{3}: 1$
(c) $1: 2: 3$
(d) $2: 1: 6$

Key Concepts

Arithmetic Progression

An arithmetic progression (AP) is a sequence of numbers in which each term after the first is obtained by adding a fixed constant called the common difference to the preceding term. Understanding AP is crucial because it defines a linear relationship between consecutive elements in a sequence, and this property is often used to express one term in terms of another, facilitating the setup of equations when combined with additional conditions.

Geometric Progression

A geometric progression (GP) is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant known as the common ratio. This multiplicative pattern contrasts with the additive nature of AP and is essential for problems where ratios of differences or terms are involved, helping to establish proportional relationships between elements of the sequence.

Interrelation of Sequence Properties

Many problems involve sequences that are simultaneously subject to conditions of both arithmetic and geometric progressions. This concept involves forming equations based on the properties of an AP (constant difference) and a GP (constant ratio), then solving the system of equations to find relationships or ratios among the terms. It requires a solid understanding of how additive and multiplicative patterns can interact, leading to the determination of specific ratios among the terms.

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