If `a_0,a_1,a_2,……a_n` be the successive coefficients in the expnsion of `(1+x)^n` show that `(a_0-a_2+a_4……..)^2+(a_1-a_3+a_5………)^2=a_0+a_1+a_2+………..+a_n=2^n`

If a_0,a_1,a_2,.... be the coefficients in the expansion of (1+x+x^2)^n in ascending powers of x. prove that : (i) a_0a_1-a_1a_2+a_2a_3-....=0

If a_1,a_2, a_3, a_4 be the coefficient of four consecutive terms in the expansion of (1+x)^n , then prove that: (a_1)/(a_1+a_2)+(a_3)/(a_3+a_4)=(2a_2)/(a_2+a_3)dot

If a_1,a_2, a_3, a_4 be the coefficient of four consecutive terms in the expansion of (1+x)^n , then prove that: (a_1)/(a_1+a_2)+(a_3)/(a_3+a_4)=(2a_2)/(a_2+a_3)dot

If a_r is the coefficient of x^r in the expansion of (1+x)^n then a_1/a_0 + 2.a_2/a_1 + 3.a_3/a_2 + …..+n.(a_n)/(a_(n-1)) =

If a_0, a_1 , a_2 …..a_n are binomial coefficients then (1 + a_1/a_0)(1 +a_2/a_1) …………….(1 + (a_n)/(a_(n-1)) ) =

If a_1,a_2,……….,a_n are in H.P. then expression a1a2+a2a3+......+a_n-1an

If a_0, a_1, a_2, a_3 are all the positive, then 4a_0x^3+3a_1x^2+2a_2x+a_3=0 has least one root in (-1,0) if (a) a_0+a_2=a_1+a_3 and 4a_0+2a_2>3a_1+a_3 (b) 4a_0+2a_2<3a_1+a_3 (c) 4a_0+2a_2=3a_1+a_0 and 4a_0+a_2lta_1+a_3 (d) none of these

If a_1,a_2,a_3,…………..a_n are in A.P. whose common difference is d, show tht sum_2^ntan^-1 d/(1+a_(n-1)a_n)= tan^-1 ((a_n-a_1)/(1+a_na_1))

If (3 + 7x - 9x^2)^n = a_0 +a_1x + a_2 x^2 + ……+a_(2n)x^(2n) prove the a_0 +a_1 +a_2 + ……+a_(2n) = 1

If a_1,a_2 …. a_n are positive real numbers whose product is a fixed real number c, then the minimum value of 1+a_1 +a_2 +….. + a_(n-1) + a_n is :