If `a_(n+1)= (1)/(1-a_(n))` for `n gt=1` and `a_(3) = a_(1)` then find the
value of `(a_(2001))^(2001)` .

Let {a_(n)} (n gt= 1 ) be a sequence such that a_(1) = 1 and 3a_(n+1)-3a_(n)=1 for all n gt= 1 . Then find the value of a_(2002) .

If a_1 = 4 and a_(n+1) = a_(n) + 4n for n ge 1 , then the value of a_(100) is

Let a_(1), a_(2),a_(3), cdots , a_(4001) are in A.P. such that (1)/(a_(1)a_(2))+(1)/(a_(2)a_(3))+ cdots + (1)/(a_(4000) a_(4001))= 10 and a_(2)+ a_(4000) = 50 then find the value of |a_(1)-a_(4001)| .

If a_(1), a_(2), a_(3),….,a_(n) are in AP and a_(1) = 0 , then the value of ((a_(3))/(a_(2)) + (a_(4))/(a_(3)) +...+(a_(n))/a_(n-1))-a_(2) ((1)/(a_(2)) + (1)/(a_(3))+...+(1)/(a_(n-2))) is equal to a) (n-2) + (1)/((n-2)) b) (1)/((n-2)) c)(n - 2) d)(n - 1)

Three distinct A.P. s' have same first term common differences as d_(1) ,d_(2) ,d_(3) and n^(th) terms as a_(n) , b_(n) ,c_(n) respectively such that (a_(1))/(d_(1))=(2b_(1))/(d_(2))=(3c_(1))/(d_(3)). If (a_(7))/(c_(6))=(3)/(7) then find the value of (b_(7))/(c_(6)) .

Let a_(1) ,a_(2),cdots , a_(n) be an A.P. with common difference pi//6 and assume sec a_(1) sec a_(2)+sec a_(2) sec a_(3) + cdots + sec a_(n-1) sec a_(n)=k ( tan a_(n) - "tan" a_(1) ) Then find the value of k.

If a_(1), a_(2), a_(3),…….,a_(20) are in AP and a_(1) + a_(20) = 45 , then a_(1) + a_(2) + a_(3)+……+a_(20) is equal to

If (1+x+2x^(2))^(20) =a_(0)+a_(1)x+a_(2)x^(2) +a_(3)x^(3)+….+a_(40)x^(40), then find the value of a_(1)+a_(3)+a_(5)+….+a_(39) .

If (1+x-2x^(2))^(6)=1+a_(1)x+a_(2)x^(2)+….+a_(12)^(x^(12)) , then find the value of a_(2)+a_(4)+a_(6)+….+a_(12) .