Let a₁,a₂,a₃,…,aₙ be an arithmetic progression with common difference d where d0. Then the value of
tan (tan⁻¹ ((d)/(1+a₁a₂))+tan⁻¹ ((d)/(1+a₂a₃))+⋯+tan⁻¹ ((d)/(1+aₙ₋₁aₙ)))
is:

Question

Let a₁,a₂,a₃,…,aₙ be an arithmetic progression with common difference d where d>0. Then the value of
tan (tan⁻¹ ((d)/(1+a₁a₂))+tan⁻¹ ((d)/(1+a₂a₃))+⋯+tan⁻¹ ((d)/(1+aₙ₋₁aₙ)))
is:

Accepted Answer

((n-1)d)/(1+a₁aₙ). The correct option is derived from the properties of the tangent addition formula and the structure of the terms in the series. Each term in the sum represents an angle whose tangent can be expressed in terms of the arithmetic progression, leading to a cumulative result that simplifies to ((n-1)d)/(1+a₁aₙ), reflecting the relationship between the first and last terms of the sequence.

Let a₁,a₂,a₃,…,aₙ be an arithmetic progression with common difference d where d>0. Then the value of tan (tan⁻¹ ((d)/(1+a₁a₂))+tan⁻¹ ((d)/(1+a₂a₃))+⋯+tan⁻¹ ((d)/(1+aₙ₋₁aₙ))) is:

Solution

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