An A.P., a G.P and a H.P. have a and b for their first two...

An A.P., a G.P and a H.P. have $a$ and $b$ for their first two terms. Show that their $(n+2)$ th terms will be in G.P. if $\frac{b^{2 n+2}-a^{2 n+2}}{a b\left(b^{2 n}-a^{2 n}\right)}=\frac{n+1}{n}$.

Key Concepts

Arithmetic Progression

An arithmetic progression is a sequence of numbers in which the difference between consecutive terms is constant. This constant difference is known as the common difference, and the nth term is typically expressed as the first term plus (n-1) times the common difference. Understanding this linear structure is essential when comparing or combining it with other types of sequences.

Geometric Progression

A geometric progression is a sequence where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. The nth term of a geometric progression is given by the first term multiplied by the common ratio raised to the power of (n-1). This exponential structure plays a key role in identifying and testing conditions under which modified terms of different sequences fall into a geometric progression.

Harmonic Progression

A harmonic progression is a sequence of numbers where the reciprocals of the terms form an arithmetic progression. This means that while the original sequence does not show a constant difference, its transformed reciprocal sequence does. Recognizing this relationship allows one to derive properties and term formulas of the harmonic progression by studying the corresponding arithmetic progression.

Conditional Relationships in Sequences

In problems that bridge multiple types of progressions, such as arithmetic, geometric, and harmonic progressions, it is often necessary to identify conditions under which certain terms from these sequences align to form another specific sequence, like a geometric progression. This involves algebraic manipulation and comparing expressions derived from the formulae for the nth terms of the different progressions, leading to conditions or equations that must be satisfied for the transformation to hold.

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