Let `(a_0)/(n+1)+(a_1)/n+(a_2)/(n-1)++(a_(n-1))/2+a_n=0.` Show that there exists at least one real `x` between 0 and 1 such that `a_0x^n+a_1x^(n-1)+a_2x^(n-2)++a_n=0`

Prove, Using the Rolle's theorem that between any two distinct real zeros of the polynomial a_(n) x^(n) + a_(n-1) x^(n-1) + …..a_(1)x + a_(0) there is a zero of the polynomial . na_(n) x^(n-1) + (n-1) a_(n-1) x^(n-2) + ….+ a_(1)

Write the first three terms of the sequence defined by a_1 2,a_(n+1)=(2a_n+3)/(a_n+2) .

Let (1 + x)^(n) = 1 + a_(1)x + a_(2)x^(2) + ... + a_(n)x^(n) . If a_(1),a_(2) and a_(3) are in A.P., then the value of n is

Find the first five terms of the following sequence . a_1 = 1 , a_2 = 1 , a_n = (a_(n-1))/(a_(n-2) + 3) , n ge 3 , n in N

Let nge1, n in Z . The real number a in 0,1 that minimizes the integral int_(0)^(1)|x^(n)-a^(n)|dx is

If a_1=1a n da_(n+1)=(4+3a_n)/(3+2a_n),ngeq1,a n dif("lim")_(nvecoo)a_n=a , then find the value of adot

Find the first 6 terms of the sequence given by a_1=1, a_n=a_(n-1)+2 , n ge 2

Let a_(0)=0 and a_(n)=3a_(n-1)+1 for n ge 1 . Then the remainder obtained dividing a_(2010) by 11 is

Find the coefficient of x^n in the polynomial (x+^n C_0)(x+3^n C_1)xx(x+5^n C_2)[x+(2n+1)^n C_n]dot

If 1/(sqrt(4x+1)){((1+sqrt(4x+1))/2)^n-((1-sqrt(4x+1))/2)^n}=a_0+a_1x then find the possible value of ndot