If `a_(1)=1` and `a_(n+1)=(4+3a_(n))/(3+2a_(n)),nge1`, show that `a_(n+2)gea_(n+1)` and if a lim l as `n to oo` the evaluate `lim_(ntooo)a_(n)``

evaluate lim_(n->oo)((e^n)/pi)^(1/ n)

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 .

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

If x_1=sqrt(3) and x_(n+1)=(x_n)/(1+sqrt(1+x_ n^2)),AA n in N then lim_(n->oo)2^n x_n is equal to

Let (1 + x^(2))^(2) (1 + x)^(n) = a_(0) + a_(1) x + a_(2) x^(2) + … if a_(0),a_(1) " and " a_(2) are in A.P , the value of n is

Consider (1 + x + x^(2))^(n) = sum_(r=0)^(n) a_(r) x^(r) , where a_(0), a_(1), a_(2),…, a_(2n) are real number and n is positive integer. The value of sum_(r=0)^(n-1) a_(r) is

Let a_(1),a_(2),a_(n) be sequence of real numbers with a_(n+1)=a_(n)+sqrt(1+a_(n)^(2)) and a_(0)=0 . Prove that lim_(xtooo)((a_(n))/(2^(n-1)))=2/(pi)

Consider (1 + x + x^(2))^(n) = sum_(r=0)^(n) a_(r) x^(r) , where a_(0), a_(1), a_(2),…, a_(2n) are real number and n is positive integer. If n is odd , the value of sum_(r-1)^(2) a_(2r -1) is

Consider (1 + x + x^(2))^(n) = sum_(r=0)^(2n) a_(r) x^(r) , where a_(0), a_(1), a_(2),…, a_(2n) are real number and n is positive integer. If n is even, the value of sum_(r=0)^(n/2-1) a_(2r) is

Let a_(0)=2,a_1=5 and for n ge 2, a_n=5a_(n-1)-6a_(n-2) . Then prove by induction that a_(n)=2^(n)+3^(n) forall n in Z^+ .