Discrete Mathematics with Applications 4th Edition

Published by Cengage Learning
ISBN 10: 0-49539-132-8
ISBN 13: 978-0-49539-132-6

Chapter 5 - Sequences, Mathematical Induction, and Recursion - Exercise Set 5.4 - Page 277: 4

Answer

See below.

Work Step by Step

1. Let $P(n)$ be the statement to be proved. 2. Test for $n=1,2,3$, we have $d_1=\frac{9}{10},d_2=\frac{10}{11},d_3=\frac{9}{10}\cdot\frac{10}{11}=\frac{9}{11}$ all are within $(0,1]$, thus $ P(1),P(2),P(3)$ are true. 3. Suppose it is true for $n\le p, (p\gt3)$. 4. For $n=p+1$, we have $d_{p+1}=d_p\cdot d_{p-1}\le (1)(1)=1$ and it is within $(0,1]$.. 5. Thus $P(p+1)$ will also be true and we have proved the statement using mathematical induction.
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