(9)+2+3+cdots+n<(1)/(8)(2m+1)^(2)

Find the sum 1+(1)/(1+2) + (1)/(1+2+3) + cdots+ (1)/(1+2+3+cdots+ n )

The mean of the squares of the numbers 1,2,3,4, cdots n is

Let P(x) =a_(0)+a_(1)x^2+a_(2)x^(4)+cdots+a_(n)x^(2n) with all a_(i) gt0 , i=0,1,2,3, cdots ,n . Then P(x) has

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

Find the sum of the series 1.n + 2 .(n-1) + 3.(n-2) + cdots + ( n-1) .2 + n.1.

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)

The sum of (x+2)^(n-1)+ (x+2)^(n-2) (x+1) + (x+2)^(n-3) (x+1)^(2) + cdots + (x+1)^(n-1) is equal to