If a1, a2,..... an is an AP with common difference d, then tan [tan-1 d / [1 + a1a2] + tan-1 d / [1 + a2a3] + …. + tan-1 d / [1 + an-1an]] =

1) [n – 1] d / [a1 + an]

2) [n – 1] d / [1 + a1an]

3) nd / [1 + a1an]

4) [an – a1] / [an + a1]

Solution: (2) [n – 1] d / [1 + a1an]

tan-1 d / [1 + a1a2] + tan-1 d / [1 + a2a3] + …. Tan-1 d / [1 + an-1an]

= tan-1 [a2 – a1] / [1 + a1a2] + tan-1 [a3 – a2] / [1 + a2a3] + …. + tan-1 [an – an-1] / [1 + an-1an]

= tan-1 a2 – tan-1 a1 + tan-1 a3 – tan-1 a2 + …. + tan-1 an – tan-1 an-1 +

= tan-1 an – tan-1 a1

= tan-1 [an – a1] / [1 + ana1]

= tan-1 [(n – 1) d] / [1 + a1an]

= tan[tan-1 [(n – 1) d] / [1 + a1an]]

= [(n – 1) d] / [1 + a1an]

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