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.

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- May 13th 2010, 08:02 PMMathman87Proof by induction
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.

- May 13th 2010, 09:03 PMsimplependulum

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 .