Let the sequence `a_(1) , a_(2) ,a_(3) ,cdots, a_(n)` form an A.P ., then prove that
`a_(1)^(2) -a_(2)^(2)+a_(3)^(2) -a_(4)^(2)+cdots+a_(2n-1)^(2) -a_(2n)^(2)= (n)/(2n-1)(a_(1)^(2)-a_(2n)^(2))`

If a_(1), a_(2),a_(3), cdots , a_(2n+1) are in A.P then (a_(2n+1)-a_(1))/(a_(2n+1)+a_(1)) + (a_(2n)-a_(2))/(a_(2n)+a_(2))+cdots+ (a_(n+_2)-a_(n))/(a_(n+2)+a_(n)) is equal to

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

If a , a_(1) , a_(2) ,a_(3) , cdots a_(2n), b are in A.P. and a, g_(1) , g_(2) , g_(3) , cdots , g_(2n), b are in G.P. and h is the H.M of a and b then prove that (a_(1)+a_(2n))/(g_(1)g_(2n))+(a_(2)+a_(2n-1))/(g_(2)g_(2n-1))+cdots+ (a_(n)+a_(n+1))/(g_(n)g_(n+1))=(2n)/(h)

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_(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.

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)

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_(50) are in GP, then " "(a_(1)-a_(3)+a_(5)-…+a_(49))/(a_(2)-a_(4)+a_(6)-…+a_(50)) is equal to :

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