If `a_(1),a_(2),a_(3),…a_(n+1)` are in arithmetic progression, then `sum_(k=0)^(n) .^(n)C_(k.a_(k+1)` is equal to (a)`2^(n)(a_(1)+a_(n+1))` (b)`2^(n-1)(a_(1)+a_(n+1))` (c)`2^(n+1)(a_(1)+a_(n+1))` (d)`(a_(1)+a_(n+1))`

If A_(1), A_(2),..,A_(n) are any n events, then

If (a_(1)+ib_(1))(a_(2)+ib_(2))………………(a_(n)+ib_(n))=A+iB , then sum_(i=1)^(n) tan^(-1)(b_(i)/a_(i)) is equal to

If a_(1),a_(2),a_(3),".....",a_(n) are in HP, than prove that a_(1)a_(2)+a_(2)a_(3)+a_(3)a_(4)+"....."+a_(n-1)a_(n)=(n-1)a_(1)a_(n)

If a_1,a_2,a_3,... are in A.P. and a_i>0 for each i, then sum_(i=1)^n n/(a_(i+1)^(2/3)+a_(i+1)^(1/3)a_i^(1/3)+a_i^(2/3)) is equal to (a) (n)/(a_(n)^(2//3)+a_(n)^(1//3)+a_(1)^(2//3)) (b) (n(n+1))/(a_(n)^(2//3)+a_(n)^(1//3)+a_(1)^(2//3)) (c) (n(n-1))/(a_(n)^(2//3)+a_(n)^(1//3)*a_(1)^(1//3)+a_(1)^(2//3)) (d) None of these

If the sequence a_(1),a_(2),a_(3),…,a_(n) is an A.P., then prove that a_(1)^(2)-a_(2)^(2)+a_(3)^(2)-a_(4)^(2)+…+a_(2n-1)^(2)-a_(2n)^(2)=n/(2n-1)(a_(1)^(2)-a_(2n)^(2))

If a_(1)=3 and a_(n)=2a_(n-1)+5 , find a_(4) .

If a_(1),a_(2),a_(3),"........",a_(n) are in AP with a_(1)=0 , prove that (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)))=(a_(n-1))/(a_(2))+(a_(2))/(a_(n-1)) .

If a_(1),a_(2),a_(3)"....." are in GP with first term a and common ratio r, then (a_(1)a_(2))/(a_(1)^(2)-a_(2)^(2))+(a_(2)a_(3))/(a_(2)^(2)-a_(3)^(2))+(a_(3)a_(4))/(a_(3)^(2)-a_(4)^(2))+"....."+(a_(n-1)a_(n))/(a_(n-1)^(2)-a_(n)^(2)) is equal to

If a_(a), a _(2), a _(3),…., a_(n) are in H.P. and f (k)=sum _(r =1) ^(n) a_(r)-a_(k) then (a_(1))/(f(1)), (a_(2))/(f (2)), (a_(3))/(f (n)) are in :

If a_(a), a _(2), a _(3),…., a_(n) are in H.P. and f (k)=sum _(r =1) ^(n) a_(r)-a_(k) then (a_(1))/(f(1)), (a_(2))/(f (2)), (a_(3))/(f (n)) are in :