Prove that `1^2+2^2+""dot""""dot""""dot""+n^2>(n^3)/3,""` `n in N`

Prove that [(n+1)//2]^n >(n !)dot

Prove that [(n+1)//2]^n >(n !)dot

Given that bar x is the mean and sigma^2 is the variance of n observations x_1x_2 , ..., x_n . Prove that the mean and variance of the observations a x_1,""""a x_2 , a x_3,""""dot""""dot""""dot"""",""a x_n are a bar x and a^2sigma^2 ,

Prove the following by using the principle of mathematical induction for all n in N : 1^3+2^3+3^3+""dot""""dot""""dot+n^3=((n(n+1))/2)^2

Let a_1,""""a_2,""""dot""""dot""""dot"",""a_n be fixed real numbers and define a function (x-a_2)""dot""""dot""""dot""(x-a_n) . What is (lim)_(x->x_1)(x) ? For some a!=a_1,""""a_2,""""dot""""dot""""dot,""""a_n , compute (lim)_(x->a)f(x)

Prove that (1^2)/3""^n C_1+(1^2+2^2)/5""^n C_2+(1^1+2^2+3^2)/7""^n C_3+... +(1^2+2^2++n^2)/(2n+1)""^n C_n = ((n(n+3))/6)*2^(n-2)dot

Find the sum to n terms of the series : 3xxa^2+5xx2^2+7xx3^2+""dot""""dot""""dot

Prove that : 1^3+2^3+3^3++n^3={(n(n+1))/2}^2dot

Show that (1xx2^2+2xx3^2+dotdotdot+nxx(n+1)^2)/(1^2xx2+2^2xx3+dotdotdot+n^2xx(n+1))=(3n+5)/(3n+1)dot

Show that (1xx2^2+2xx3^2+dotdotdot+nxx(n+1)^2)/(1^2xx2+2^2xx3+dotdotdot+n^2xx(n+1))=(3n+5)/(3n+1)dot