If `b_1, b_2 b_n` are the nth roots of unity, then prove that `^n C_1dotb_1+^n C_2dotb_2++^n C_ndotb_n-b_1/b_2{(1+b_2)^n-1}^dot`

If a > b and n is a positive integer, then prove that a^n-b^n > n(a b)^((n-1)//2)(a-b)dot

If (1+x)^n=sum_(r=0)^n C_r x^r , then prove that C_1+2C_2+3C_3+....+n C_n=n2^(n-1)dot .

If omega be a nth root of unity, then 1+omega+omega^2+…..+omega^(n-1) is (a)0(B) 1 (C) -1 (D) 2

If 3t a n At a n B=1, prove that 2cos(A+B)=cos(A-B)dot

A B C is a right-angled triangle in which /_B=90^0 and B C=adot If n points L_1, L_2, ,L_nonA B is divided in n+1 equal parts and L_1M_1, L_2M_2, ,L_n M_n are line segments parallel to B Ca n dM_1, M_2, ,M_n are on A C , then the sum of the lengths of L_1M_1, L_2M_2, ,L_n M_n is (a(n+1))/2 b. (a(n-1))/2 c. (a n)/2 d. none of these

If n. sin(A+2B)=sinA , then prove that: tan(A+B)=(1+n)/(1-n).tanB

If alpha_(1), alpha_(2), …….., alpha_(n) are the n,n^(th) roots of unity, alpha_(r )=e (i2(r-1)pi)/(n), r=1,2,…n then ""^(n)C_(1)alpha_(1)+""^(n)C_(2)alpha_(2)+…..+""^(n)C_(n)alpha_(n) is equal to :

If f(x)=(a_1x+b_1)^2+(a_2x+b_2)^2+...+(a_n x+b_n)^2 , then prove that (a_1b_1+a_2b_2+...+a_n b_n)^2lt=(a_1^ 2+a_2^ 2+...+a_ n^2)(b_1^ 2+b_2^ 2+..+b_ n^2)dot

If 1,alpha_1, alpha_2, …alpha_(n-1) be nth roots of unity then (1+alpha_1)(1+alpha_2)……....(1+alpha_(n-1))= (A) 0 or 1 according as n is even or odd (B) 0 or 1 according as n is odd or even (C) n (D) -n

If alpha is the nth root of unity, then 1+2alpha+3alpha^2+ ton terms equal to a. (-n)/((1-alpha)^2) b. (-n)/(1-alpha) c. (-2n)/(1-alpha)"" d. (-2n)/((1-alpha)^2)