Let `n` be a positive integer and `(1+x+x^2)^n=a_0+a_1x++a^(2n)x^(2n)dot` Show that `a0 2-a1 2+a2 2++a2n2=a_ndot`

Let n be a positive integer, such that (1 + x + x^2)^n=a_0 +a_1x+ a_2 x^2+a_3 x^3+......a_(2n) x^(2n), Answer the following question: The value of a when (0 leq r leq n) is (A) a_(2n-r) (B) a_(n-r) (C) a_(2n) (D) n. a_(2n-1)

If (1+x-x^2)^n/(1+x^2)=a_0+a_1x+a_2x^2+...+a_(2n)x^(2n) then find a_0-a_1+a_2_...+a_(2n)

If (1+x-x^2)^n/(1+x^2)=a_0+a_1x+a_2x^2+...+a_(2n)x^(2n) then find a_0+a_1+a_2+...+a_(2n)

Let f_1 : (0, oo) to R and f_2 : (0, oo) to R be defined by f_1(x) = int_0^xprod_(j= 1)^21((t − j)^j)dt, x gt 0 and f_2(x) = 98(x − 1)^50 − 600(x − 1)^49 + 2450, x gt 0 , where, for any positive integer n and real numbers a_1, a_2, … , a_n, prod_(i=1)^n(a_i) denotes the product of a_1 , a_2, … , a_n . Let m_i and n_i , respectively, denote the number of points of local minima and the number of points of local maxima of function f_i, i = 1, 2 , in the interval (0, oo) The value of 6m_2+4n_2+8m_2n_2 is ___.

If n is a positive integer and if (1+x+x^(2))^(2n)=underset(r=0)overset(2n)(sum)a_(r)x^(r) , then

Let f_1 : (0, oo) to R and f_2 : (0, oo) to R be defined by f_1(x) = int_0^xprod_(j= 1)^21((t − j)^j)dt, x gt 0 and f_2(x) = 98(x − 1)^50 − 600(x − 1)^49 + 2450, x gt 0 , where, for any positive integer n and real numbers a_1, a_2, … , a_n, prod_(i=1)^n(a_i) denotes the product of a_1 , a_2, … , a_n . Let m_i and n_i , respectively, denote the number of points of local minima and the number of points of local maxima of function f_i, i = 1, 2 , in the interval (0, oo) The value of 2m_1+3n_1+m_1n_1 is ___.

Given that (1+x+x^2)^n=a_0+a_1x+a_2x^2+.....+a_(2n)x^(2n) find i) a_0 + a_1 +a_2 .. . . .+ a_(2n) ii) a_0 - a_1 + a_2 - a_3 . . . . + a_(2n) iii) (a_0)^2 - (a_1)^2 . . . . .+ (a_(2n))^2

If (1+2x-x^2)^n/(1+x^2)=a_0+a_1x+a_2x^2+...+a_(2n)x^(2n) then find a_0+a_2+a_4+...+a_(2n)