Confused by Induction Proof Problem

sritter27 · #1 $Old$ January 22nd, 2009, 10:48 PM

I have to admit that I confuse myself very easily with even the simplest of proofs that require inductive reasoning, so hopefully somebody can help me out with the following question and point me in the right direction.

For a positive integer n the number a_n is defined inductively by
a_1 = 1
a_k+1 = (6a_k + 5)/(a_k + 2) for k a positive integer
Prove by induction on n that, for all positive integers, a_n > 0 and a_n < 5.

This is problem 17 in Part I of "An Introduction to Mathematical Reasoning" by Peter J. Eccles if anyone cares and any help will be greatly appreciated.

Opalg · #2 $Old$ January 24th, 2009, 12:16 PM

Quote:

Originally Posted by sritter27 $View Post$

I have to admit that I confuse myself very easily with even the simplest of proofs that require inductive reasoning, so hopefully somebody can help me out with the following question and point me in the right direction.

For a positive integer n the number a_n is defined inductively by
a_1 = 1
a_k+1 = (6a_k + 5)/(a_k + 2) for k a positive integer
Prove by induction on n that, for all positive integers, a_n > 0 and a_n < 5.

This is problem 17 in Part I of "An Introduction to Mathematical Reasoning" by Peter J. Eccles if anyone cares and any help will be greatly appreciated.

Write a_{k+1} as $a_{k+1} = 6 - \frac7{a_k+2}$ . If $0<a_k<5$ then $2<a_k+2<7$ , so $\frac77< \frac7{a_k+2} < \frac72$ ... .