Let a1=3 and a(n+1) =( )/7 for every n 1
Prove by induction that 2 an 5 for every n 1
Deduce that an+1 an for every n 1
Show the sequence {an} converges and determine its limit.
Induction on , and assume it is true that and we have
Also ,
Therefore ,
Hence we deduce it is true that
Consider
because
but from
Therefore , is a non-positive number or
Note that the sequence is bounded and it is monotonic decreasing , use some theorems to show that the limit exists and find it .