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4. (a) Show that if n is any integer, then precisely one of n – 1, n, and n+1 is divisible by 3. Hint. Use the division algorithm. (b) For each n e N, show that Fn is even if and only if 3 | n, where F, is the nth Fibonacci number. Hint. In your inductive step to prove the result for n+1, break it into two cases: n+1 is divisible by 3 and n + 1 is not divisible by 3.

4. (a) Show that if n is any integer, then precisely one of n – 1, n, and n+1 is divisible by 3. Hint. Use the division algorithm. (b) For each n e N, show that Fn is even if and only if 3 | n, where F, is the nth Fibonacci number. Hint. In your inductive step to prove the result for n+1, break it into two cases: n+1 is divisible by 3 and n + 1 is not divisible by 3.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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4. (a) Show that if n is any integer, then precisely one ofn – 1, n, and n +1 is divisible by 3. Hint. Use the division algorithm. (b) For each n e N, show that Fn is even if and only if 3 | n, where F, is the nth Fibonacci number. Hint. In your inductive step to prove the result for n+1, break it into two cases: n+1 is divisible by 3 and n+1 is not divisible by 3.
Transcribed Image Text:4. (a) Show that if n is any integer, then precisely one ofn – 1, n, and n +1 is divisible by 3. Hint. Use the division algorithm. (b) For each n e N, show that Fn is even if and only if 3 | n, where F, is the nth Fibonacci number. Hint. In your inductive step to prove the result for n+1, break it into two cases: n+1 is divisible by 3 and n+1 is not divisible by 3.
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