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Write out careful proofs, or give counterexamples, to the following statements. (a) Show that the sum of the squares of any three consecutive integers, plus one, is divisible by 3.

Write out careful proofs, or give counterexamples, to the following statements. (a) Show that the sum of the squares of any three consecutive integers, plus one, is 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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Write out careful proofs, or give counterexamples, to the following statements. (a) Show that the sum of the squares of any three consecutive integers, plus one, is divisible by 3. (b) Show that the sum of any four consecutive integers is divisible by 4. (c) A function ƒ: A → B has an inverse if and only if it is both surjective and injective. (d) If f is not surjective, then it is injective. (e) If g is surjective, then fog is surjective. (f) If g and fog are injective, then 9 is injective. (g) If ƒ og: R → R is decreasing, and g: R → R is decreasing, then f is decreasing. (h) If x = R and x² < x, then x < 1.
Transcribed Image Text:Write out careful proofs, or give counterexamples, to the following statements. (a) Show that the sum of the squares of any three consecutive integers, plus one, is divisible by 3. (b) Show that the sum of any four consecutive integers is divisible by 4. (c) A function ƒ: A → B has an inverse if and only if it is both surjective and injective. (d) If f is not surjective, then it is injective. (e) If g is surjective, then fog is surjective. (f) If g and fog are injective, then 9 is injective. (g) If ƒ og: R → R is decreasing, and g: R → R is decreasing, then f is decreasing. (h) If x = R and x² < x, then x < 1.
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