If `a_1, a_2, ,a_n >0,` then prove that `(a_1)/(a_2)+(a_2)/(a_3)+(a_3)/(a_4)++(a_(n-1))/(a_n)+(a_n)/(a_1)> n`

If a_1,a_2,….a_n are in H.P then (a_1)/(a_2+,a_3,…,a_n),(a_2)/(a_1+a_3+….+a_n),…,(a_n)/(a_1+a_2+….+a_(n-1)) are in

If a ,a_1, a_2, a_3, a_(2n),b are in A.P. and a ,g_1,g_2,g_3, ,g_(2n),b . are in G.P. and h s the H.M. of aa n db , then prove that (a_1+a_(2n))/(g_1g_(2n))+(a_2+a_(2n-1))/(g_1g_(2n-1))++(a_n+a_(n+1))/(g_ng_(n+1))=(2n)/h

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_(2n+1) are in A.P then (a_(2n+1)-a_1)/(a_(2n+1)+a_1)+(a_2n-a_2)/(a_(2n)+a_2)+....+(a_(n+2)-a_n)/(a_(n+2)+a_n) is equal to

If a_1,a_2,a_3, ,a_n are in A.P., where a_i >0 for all i , show that 1/(sqrt(a_1)+sqrt(a_2))+1/(sqrt(a_1)+sqrt(a_3))++1/(sqrt(a_(n-1))+sqrt(a_n))=(n-1)/(sqrt(a_1)+sqrt(a_n))dot

If S+a_1+a_2+a_3++a_n ,a_1 in R^+ for i=1ton , then prove that S/(S-a_1)+S/(S-a_2)++S/(S-a_n)geq(n^2)/(n-1),AAngeq2

If roots of an equation x^n-1=0 are 1,a_1,a_2,........,a_(n-1) then the value of (1-a_1)(1-a_2)(1-a_3)........(1-a_(n-1)) will be (a) n (b) n^2 (c) n^n (d) 0

If a_1,a_2,a_3 ,….""a_n is an aritmetic progression with common difference d. prove that tan [ tan ^(-1) ((d)/( 1+ a_1a_2))+ tan^(-1) ((d)/(1+a_2a_3))+…+((d)/(1+a_n a_( n-1))) ] = (a_n -a_1)/(1 +a_1 a_n )

If a_1,a_2,a_3…a_n are in H.P and f(k)=(Sigma_(r=1)^(n) a_r)-a_k then a_1/f(1),a_2/f(3),….,a_n/f(n) are in

Let vec a=a_1 hat i+a_2 hat j+a_3 hat k , vec b=b_1 hat i+b_2 hat j+b_3 hat ka n d vec c=c_1 hat i+c_2 hat j+c_3 hat k be three non-zero vectors such that vec c is a unit vector perpendicular to both vec aa n d vec b . If the angle between aa n db is pi/6, then prove that |(a_1 a_2a_3)(b_1b_2b_3)(c_1c_2c_3)|=1/4(a1 2+a2 2+a3 2)(b1 2+b2 2+b3 2)