For each positive integer `n gt a,A_(n)` represents the area of the region restricted to the following two inequalities : `(x ^(2))/(n ^(2)) + y ^(2) and x ^(2)+ (y ^(2))/(n ^(2)) lt 1.` Find `lim _(n to oo) A_(n).`

For each positive integer n, let A_(n)=max{(C(n,r)0<=r<=n} then the number of elements n in {1,2...20} for 1.9<=(A_(n))/(A_(n-1))<=2 is

Find the indicated terms in each of the following sequences whose n^(th) terms are a_(n) = (a_(n-1))/(n^(2)) , (n ge 2) , a_(1) =3 , a_(2) , a_(3)

Find the indicated terms in each of the following sequences whose n^(th) terms are a_(n) = (a_(n-1))/(n^(2)) , (n ge 2) , a_(1) =3 , a_(2) , a_(3)

Let n be positive integer such that, (1+x+x^(2))^(n)=a_(0)+a_(1)x+a_(2)x^(2)+….+a_(2n)x^(2n) , then a_(r) is :

The number of real roots of : 1 + a_(1)X + a_(2) x^(2) .... + a_(n) x^(n) = 0 , where |x| lt (1)/(3) and | a_(n) | lt 2 , is :

Let A_(n) be the area bounded by the curve y=x^(n)(n>=1) and the line x=0,y=0 and x=(1)/(2). If sum_(n=1)^(n)(2^(n)A_(n))/(n)=(1)/(3) then find the value of n .

If (1+x+x^(2))^(n) = sum_(r=0)^(2 n) a_(r). x^(r) then a_(1)-2 a_(2)+3 a_(3)-..-2 n. a_(2n) =

Let n be a positive integer, such that (1 + x + x^2)^n=a_0 +a_1x+ a_2 x^2+a_3 x^3+......a_(2n) x^(2n), Answer the following question: The value of a when (0 leq r leq n) is (A) a_(2n-r) (B) a_(n-r) (C) a_(2n) (D) n. a_(2n-1)

The number of real roots of the equation 1+a_(1)x+a_(2)x^(2)+………..a_(n)x^(n)=0 , where |x| lt (1)/(3) and |a_(n)| lt 2 , is