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 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 the equation z^4+a_1z^3+a_2z^2+a_3z+a_4=0 where a_1,a_2,a_3,a_4 are real coefficients different from zero has a pure imaginary root then the expression (a_3)/(a_1a_2)+(a_1a_4)/(a_2a_3) has the value equal to

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,A_1,A_2 and A_3 are the areas of the inscribed and escribed circles of a triangle, prove that 1/sqrtA=1/sqrt(A_1)+1/sqrt(A_2)+1/sqrt(A_3)

Evaluate: /_\ |[1+a_1, a_2, a_3],[a_1, 1+a_2, a_3],[a_1, a_2, 1+a_3]|

Evaluate: /_\ |[1+a_1, a_2, a_3],[a_1, 1+a_2, a_3],[a_1, a_2, 1+a_3]|

If a_1,a_2,a_3,.....,a_(n+1) be (n+1) different prime numbers, then the number of different factors (other than1) of a_1^m.a_2.a_3...a_(n+1) , is

If a_r=(cos2rpi+i sin 2rpi)^(1//9) , then prove that |[a_1,a_2,a_3],[a_4,a_5,a_6],[a_7,a_8,a_9]|=0 .

Find the sum of first 24 terms of the A.P. a_1, a_2, a_3, , if it is know that a_1+a_5+a_(10)+a_(15)+a_(20)+a_(24)=225.