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

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 i= sqrt-1 . If a sequence of complex numbers is defined by z_(1)=0, z_(n+1)= z_(n)^(2)+i for n ge 1 , then find the value of z_(111) .

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) , cdots ,a_(n) are in A.P. with a_(1) =3, a_(n) =39 and a_(1) +a_(2) + cdots +a_(n) =210 then the value of n is equal to

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) ,a_(3) ,cdots ,a_(n) are in A.P such that a_(n) =100 , a_(10)- a_(39 ) = (3)/(5) then find the 15^(th) term of A.P. from end.

Find the first 5 terms of the sequence having the property a_1=3,a_n=3a_(n-1)+2 for all ngt1 . Also obtain the corresponding series

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)| .