If f(x+y)=f(x)f(y) for all x and y,f(1)=2 and `a_(n)=f(n),n in N,` then the equation of the circle having `(a_(1),a_(2))` and `(a_(3),a_(4))` as the ends of its one diameter

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.

(1+x)^(n)=a_(0)+a_(1)x+a_(2)x^(2) +......+a_(n)x^(n) then Find the sum of the series a_(0) +a_(2)+a_(4) +……

If alpha and beta are roots of the equation x^(2)-3x+1=0 and a_(n)=alpha^(n)+beta^(n)-1 then find the value of (a_(5)-a_(1))/(a_(3)-a_(1))

Let f(x) be a polynomial function of the second degree. If f(1)=f(-1) and a_(1), a_(2),a_(3) are in AP, then f'(a_(1)),f'(a_(2)),f'(a_(3)) are in :

Let f(x) = a_(0)x^(n) + a_(1)x^(n-1) + a_(2) x^(n-2) + …. + a_(n-1)x + a_(n) , where a_(0), a_(1), a_(2),...., a_(n) are real numbers. If f(x) is divided by (ax - b), then the remainder is

If A_(n) is the area bounded by y=x and y=x^(n), n in N, then A_(2).A_(3)…A_(n)=