if `a_(r+1)=sqrt(1/2(1+a_r)`,prove that `cos(sqrt(1-a_0^2)/(a_1*a_2*a_3.....oo))=a_0`

If a_(r+1) = sqrt(1/2 1+a_r) , prove that cos ((sqrt1-a_0^(2))/(a_1*a_2*a_3*.....to ∞)) = a_0 .

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

If 1, a_1,a_2,a_3 ,…, a_(n-1) are the nth roots of unity then prove that : (1-a_1)(1-a_2)(1-a_3)...(1-a_(n-1)) =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

Given a_1=1/2(a_0+A/(a_0)), a_2=1/2(a_1+A/(a_1)) and a_(n+1)=1/2(a_n+A/(a_n)) for n>=2 , where a>0,A>0 . prove that (a_n-sqrt(A))/(a_n+sqrt(A))=((a_1-sqrt(A))/(a_1+sqrt(A)))2^(n-1) .

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_2)+sqrt(a_3))+ ..+1/(sqrt(a_(n-1))+sqrt(a_n))=(n-1)/(sqrt(a_1)+sqrt(a_n))dot

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_2)+sqrt(a_3))++1/(sqrt(a_(n-1))+sqrt(a_n))=(n-1)/(sqrt(a_1)+sqrt(a_n))dot

If roots of an equation x^n-1=0a r e1,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 n b. n^2 c. n^n d. 0

If roots of an equation x^n-1=0a r e1,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 n b. n^2 c. n^n d. 0