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Question
If a1, a2, a3,...,an is an arithmetic progression with common difference d, then evaluate the following expression.
`tan[tan^-1("d"/(1 + "a"_1 "a"_2)) + tan^-1("d"/(21 + "a"_2 "a"_3)) + tan^-1("d"/(1 + "a"_3 "a"_4)) + ... + tan^-1("d"/(1 + "a"_("n" - 1) "a""n"))]`
Solution
If a1, a2, a3, ..., an are the terms of an arithmetic progression
∴ d = a2 – a1
= a3 – a2
= a4 – a3 ....
∴ `tan[tan^-1 (("a"_2 - "a"_1)/(1 + "a"_1"a"_2)) + tan^-1 (("a"_3 - "a"_2)/(1 + "a"_2 "a"_3)) + tan^-1 (("a"_4 - "a"_3)/(1 + "a"_3 "a"_4)) + ...... + tan^-1 (("a"_"n" - "a"_("n" - 1))/(1 + "a"_("n" - 1) * "a"_"n"))]``
⇒ tan [(tan–1 a2 – tan–1 a1) + (tan–1 a3 – tan–1 a2) + (tan–1 a4 – tan–1 a3) + ... + (tan–1 an – tan–1 an – 1)] .....`[because tan^-1 (x - y)/(1 + xy) = tan^-1x - tan^-1y]`
⇒ tan [(tan–1 a2 – tan–1 a1 + tan–1 a3 – tan–1 a2 + tan–1 a4 – tan–1 a3 + ... + tan–1 an – tan–1 an – 1]
⇒ tan [tan–1 an – tan–1 a1]
⇒ `tan[tan^-1 (("a"_"n" - "a"_1)/(1 + "a"_1"a"_"n"))]`
⇒ `("a"_"n" - "a"_1)/(1 + "a"_1"a"_"n")` .....[∵ tan (tan–1x) = x]
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