let `f(x)` be a polynomial function of second degree. If `f(1)=f(-1)and a_(1),a_(2),a_(3)` are in AP, then show that `f'(a_(1)),f'(a_(2)),f'(a_(3))` are in AP.

If a_(1),a_(2),a_(3),a_(4),a_(5) are in HP, then a_(1)a_(2)+a_(2)a_(3)+a_(3)a_(4)+a_(4)a_(5) is equal to

If a_(1)=2 and a_(n)=2a_(n-1)+5 for ngt1 , the value of sum_(r=2)^(5)a_(r) is

If a_(1)=1,a_(n+1)=(1)/(n+1)a_(n),a ge1 , then prove by induction that a_(n+1)=(1)/((n+1)!)n in N .

The number of functions from f:{a_(1),a_(2),...,a_(10)} rarr {b_(1),b_(2),...,b_(5)} is

If the function f(x) is defined as : f(x) =a_(0) +a_(1)x^(2)+…..+ a_(10)x^(20) where 0 lt a_(0) lt …. < a_(10) . Then f(x) has in RR :

If a_(1),a_(2),a_(3),"........" is an arithmetic progression with common difference 1 and a_(1)+a_(2)+a_(3)+"..."+a_(98)=137 , then find the value of a_(2)+a_(4)+a_(6)+"..."+a_(98) .

If 4a^(2)+9b^(2)+16c^(2)=2(3ab+6bc+4ca) , where a,b,c are non-zero real numbers, then a,b,c are in GP. Statement 2 If (a_(1)-a_(2))^(2)+(a_(2)-a_(3))^(2)+(a_(3)-a_(1))^(2)=0 , then a_(1)=a_(2)=a_(3),AA a_(1),a_(2),a_(3) in R .

Show that for any real numbers a_(3),a_(4),a_(5),……….a_(85) , the roots of the equation a_(85)x^(85)+a_(84)x^(84)+……….+a_(3)x^(3)+3x^(2)+2x+1=0 are not real.

Equation x^(n)-1=0, ngt1, n in N," has roots "1,a_(1),a_(2),…,a_(n-1). The value of (1-a_(1))(1-a_(2))…(1-a_(n-1)) is

Let a_(1),a_(2),a_(3) be terms of an Ap. If (a_(1)+a_(2)+----+a_(p))/(a_(1)+a_(2)----a_(q))=(p^(2))/(q^(2)), p ne q, "then" (a_(6))/(a_(21)) is