Let a1, a2,...,a, be real numbers that are all contained in the interval [-7 /2, 7 /2]. Use the Pigeonhole Principle to prove that there are two distinct indices i and j such that 0
Let a1, a2,...,a, be real numbers that are all contained in the interval [-7 /2, 7 /2]. Use the Pigeonhole Principle to prove that there are two distinct indices i and j such that 0
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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Question 3:
• Let a1, a2, ..., a7 be real numbers that are all contained in the interval [-1/2, 7/2].
Use the Pigeonhole Principle to prove that there are two distinct indices i and j such
that 0 < a; – aj < T/6.
1
• Let a1, a2, ..., a7 be real numbers such that a;a; -1 for all i + j.
Prove that there are two distinct indices i and j such that
a; – aj
1
1+ a¿aj
V3
Hint: For each i, let p; be the point with coordinates (1, a;), and consider the angle
between the x-axis and the vector from the origin to p;. You learned in high school
that
tan a – tan 3
tan(a – B)
1+ tan a tan 3
11:21 PM
10/8/2020](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F38e186c9-b731-4a95-a672-769a98bb4726%2F083c9cf2-2d5a-4e62-9525-8168bf795926%2F7sxdoqj_processed.png&w=3840&q=75)
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Question 3:
• Let a1, a2, ..., a7 be real numbers that are all contained in the interval [-1/2, 7/2].
Use the Pigeonhole Principle to prove that there are two distinct indices i and j such
that 0 < a; – aj < T/6.
1
• Let a1, a2, ..., a7 be real numbers such that a;a; -1 for all i + j.
Prove that there are two distinct indices i and j such that
a; – aj
1
1+ a¿aj
V3
Hint: For each i, let p; be the point with coordinates (1, a;), and consider the angle
between the x-axis and the vector from the origin to p;. You learned in high school
that
tan a – tan 3
tan(a – B)
1+ tan a tan 3
11:21 PM
10/8/2020
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