If the (m+1)^th ,(n+1)^sh and (r+1)^t h terms of an A.P. are in G.P., m, n, r are in H.P. show

step 1 In this question, it is given that M plus 1th term, N plus 1th term and R plus 1th term of an AP

If the (m+1)^th ,(n+1)^sh and (r+1)^t h terms of an A.P....

If the $(m+1)^{\text {th }},(n+1)^{\text {sh }}$ and $(r+1)^{t h}$ terms of an A.P. are in G.P., $m, n, r$ are in H.P. show that the ratio of the common difference to the first term in the A.P. is $-\frac{2}{n}$.

Key Concepts

Arithmetic Progression

An arithmetic progression (AP) is a sequence of numbers in which the difference between consecutive terms is constant, known as the common difference. The general term is expressed as a + nd, where a is the first term and d is the common difference. This formula serves as a fundamental tool in connecting any term of the sequence to its position.

Geometric Progression

A geometric progression (GP) is a sequence in which each term after the first is obtained by multiplying the previous term by a fixed, non-zero number called the common ratio. A key property of GP is that for any three consecutive terms, the square of the middle term equals the product of the other two, which provides a method to establish relationships between non-adjacent terms.

Harmonic Progression

A harmonic progression (HP) is a sequence of numbers such that the reciprocals of the terms form an arithmetic progression. This transformation allows problems stated in terms of HP to be recast in AP form, facilitating algebraic manipulation and the establishment of conditions between terms.

Interrelation of Progressions

In problems involving multiple types of sequences, conditions from different progressions can be combined to derive relationships among their parameters. For instance, when specific terms of an AP form a GP and their positional indices are in HP, the properties of each progression interact to produce constraints on the first term and common difference, often leading to elegant algebraic results.

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