Question

Challenge to you ...
lets see who can solve this sum
Those who have already solved it i can't help but its a beautiful question....... maximum points to y'all and marking of brainlyest ​

Answer 1

Step-by-step explanation:

we know that,

tan^-1(A) - tan^-1(B) = tan^-1(A-B/(1+AB)) ....@1

=> a1,a2,a3,a4,...... are in GP

=> d = a2-a1 or d=a3-a2 or d = a4-a3 ......... or d = an - (an-1)

=> tan(tan^-1(d/(1+a1a2)) + tan^-1(d/(1+a2a3))+ ... + tan^-1(d/(1+(an-1)(an)) )

=> tan [ tan^-1{ (a2-a1)/(1+a1a2) } + tan^-1 { (a3-a2)/(1+a3a2) }+ .... + tan^-1 { (an - a(n-1) )/(1+an*an-1) } ]

=> from equation 1 we can write

=> tan[ ( tan^-1a2 - tan^-1 a1) + (tan^-1 a3 - tan^-a2) + (tan^-1 a4 - tan^-1 a3) + .... + (tan^-1 an - tan^-1 (an-1) ]

=> tan [ tan^-1 an - tan^-1 a1 ]

=> tan [ tan^-1 { (an-a1)/(1+a1*an) } ]

=> (an - a1)/(1 + a1*an)