A sequence of numbers `A_n, n=1,2,3,..` is defined as follows : `A_1=1/2` and for each `ngeq2,` `A_n=((2n-3)/(2n))A_(n-1)` , then prove that `sum_(k=1)^n A_k<1,ngeq1`

A sequence of 2xx2 matrices {M_(n)} is defined as follows M_(n)=[((1)/((2n+1)!),(1)/((2n+2)!)),(sum_(k=0)^(n)((2n+2)!)/((2k+2)!), sum_(k=0)^(n)((2n+1)!)/((2k+1)!))] then lim_(n to oo) "det". (M_(n))=lambda-e^(-1) . Find lambda .

A sequence is defined as follows : a_(a)=4, a_(n)=2a_(n-1)+1, ngt2 , find (a_(n+1))/(a_(n)) for n=1, 2, 3.

Let {A_n} be a unique sequence of positive integers satisfying the following properties: A_1 = 1, A_2 = 2, A_4 = 12, and A_(n+1) . A_(n-1) = A_n^2 pm 1 for n = 2,3,4… then , A_7 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 ge 0 , n in N .

Let the sequence a_n be defined as follows : a_1=1,""""a_n=a_(n-1)+2 for ngeq2 . Find first five terms and write corresponding series.

Let a_(n)=sum_(k=1)^(n)(1)/(k(n+1-k)), then for n>=2

a_(k)=(1)/(k(k+1)) for k=1,2,3,4,...n then (sum_(k=1)^(n)a_(k))^(2)=

If a_1,a_2,……….,a_(n+1) are in A.P. prove that sum_(k=0)^n ^nC_k.a_(k+1)=2^(n-1)(a_1+a_(n+1))

The n^(th) term, an of a sequence of numbers is given by the formula a_n = a_(n – 1) + 2n for n ge 2 and a_1 = 1 . Find an equation expressing an as a polynomial in n. Also find the sum to n terms of the sequence.

prove that sum_(k=1)^(n)k2^(-k)=2[1-2^(-n)-n*2^(-(n+1)))