How to solve this type of sequence?
1)The sequence given is: 〖(a_n)〗_(n ϵ N) , such as:
a_1= 1/3 for every n ϵ N , a_(n+1)=(2n+1)/(4n+3) a_n.
Find a_3 ?
Considering the fact that every monotone decreasing and bounded in sequence is a convergent sequence, show that the sequence 〖(a_n)〗_(n ϵ N) is a convergent sequence.
Find lim┬(n→∞)⁡〖a_n 〗

Question

How to solve this type of sequence?
1)The sequence given is:  〖(a_n)〗_(n ϵ N)     , such as:
a_1= 1/3         for every n ϵ N  ,  a_(n+1)=(2n+1)/(4n+3) a_n.
	Find   a_3  ?
	Considering the fact that every  monotone decreasing and bounded in  sequence  is a convergent sequence, show that the sequence  〖(a_n)〗_(n ϵ N) is a convergent sequence.
	Find  lim┬(n→∞)⁡〖a_n 〗

loser66 · Answer

Why do you go to a complicated way? just plug n =1,2 into the given equation and you get $a_3$

loser66 · Answer

$a_{n+1}= \dfrac{2n+1}{(4n+3)a_n}$
If n =1, then $a_{n+1}= a_{1+1}= a_2 =\dfrac{2*1+1}{(4*1+3)a_1}$

loser66 · Answer

repeat with n =2, you get a3