# Induction proofs

• Oct 14th 2009, 08:16 PM
studentmath92
Induction proofs
1. Give a sentence P(n) depending on a natural number n, such that P(1),P(2),...,P(99) are true but P(100) is false. Make the sentence as simple as possible.
2. Let <a> be a sequence satisfying a1=a2=1 and a_n = 1/2(a_n-1 + 2/(a_n-2)) for n greater than equal 2. Prove that 1 <= a_n <= 2 for all n in the natural numbers
• Oct 14th 2009, 09:32 PM
tonio
Quote:

Originally Posted by studentmath92
1. Give a sentence P(n) depending on a natural number n, such that P(1),P(2),...,P(99) are true but P(100) is false. Make the sentence as simple as possible.
2. Let <a> be a sequence satisfying a1=a2=1 and a_n = 1/2(a_n-1 + 2/(a_n-2)) for n greater than equal 2. Prove that 1 <= a_n <= 2 for all n in the natural numbers

If you gave a little more thought to this stuff I think you could come up with pretty nice answers
1) what about n < 100 ? Is this a simple enough sentence?
2) is that a_n = (1/2)(a_(n-1) + 2/(a_(n-2))? Well, show or n =1 and then assume truth for n and show for n+1.

Tonio