If `a_(1),a_(2),a_(3),…` are in `G.P.`, where `a_(i) in C` (where `C` satands for set of complex numbers) having `r` as common ratio such that `sum_(k=1)^(n)a_(2k-1)sum_(k=1)^(n)a_(2k+3) ne 0` , then the number of possible values of `r` is

If a_(1),a_(2),a_(3),…. are in A.P., then a_(p),a_(q),a_(r) are in A.P. if p,q,r are in

Let a_(n) be the nth term of an AP, if sum_(r=1)^(100)a_(2r)= alpha " and "sum_(r=1)^(100)a_(2r-1)=beta , then the common difference of the AP is

If (1+2x+x^(2))^(n) = sum_(r=0)^(2n)a_(r)x^(r) , then a_(r) =

For any n positive numbers a_(1),a_(2),…,a_(n) such that sum_(i=1)^(n) a_(i)=alpha , the least value of sum_(i=1)^(n) a_(i)^(-1) , is

If (4x^(2) + 1)^(n) = sum_(r=0)^(n)a_(r)(1+x^(2))^(n-r)x^(2r) , then the value of sum_(r=0)^(n)a_(r) is

If a_(1), a_(2), …..,a_(n) are in A.P. with common difference d ne 0, then the sum of the series sin d[sec a_(1)sec a_(2) +..... sec a_(n-1) sec a_(n)] is

Let a _(1), a_(2), a_(3),…….., a_(n) be real numbers in arithmatic progression such that a _(1) =15 and a_(2) is an integer. Given sum _( r=1) ^(10) (a_(r)) ^(2) =1185 . If S_(n) = sum _(r =1) ^(n) a_(r) and maximum value of n is N for which S_(n) ge S_((n +1)), then find N -10.

If a_(0), a_(1), a_(2),… are the coefficients in the expansion of (1 + x + x^(2))^(n) in ascending powers of x, prove that a_(0) a_(2) - a_(1) a_(3) + a_(2) a_(4) - …+ a_(2n-2) a_(2n)= a_(n+1) .

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

If a_(1),a_(2),a_(3),"….",a_(n) are in AP, where a_(i)gt0 for all I, the value of (1)/(sqrta_(1)+sqrta_(2))+(1)/(sqrta_(2)+sqrta_(3))+"....."+(1)/(sqrta_(n-1)+sqrta_(n)) is