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a. Prove that (2) is divisible by n + 1. Hint: Simplify (²2) - (2+1).

a. Prove that (2) is divisible by n + 1. Hint: Simplify (²2) - (2+1).

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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Question
a. Prove that (2n) is divisible by n + 1. Hint: Simplify (22) - (271). b. The numbers in the previous question are called the Catalan numbers. They are (2). Use this formula to recover the values 1 given by the closed formula Cn C1, C2, C3, and C₁. = = n c. Prove that the Catalan numbers satisfy the recurrence C₁+1 = n ≥ 0, and Co has the form (w₁)w2, where w₁ and we are both correct parenthetical expressions of smaller size (perhaps even empty). C₂Cn-i for i=1 1. Hint: Any correctly matched set of n + 1 pairs of parenthesis
Transcribed Image Text:a. Prove that (2n) is divisible by n + 1. Hint: Simplify (22) - (271). b. The numbers in the previous question are called the Catalan numbers. They are (2). Use this formula to recover the values 1 given by the closed formula Cn C1, C2, C3, and C₁. = = n c. Prove that the Catalan numbers satisfy the recurrence C₁+1 = n ≥ 0, and Co has the form (w₁)w2, where w₁ and we are both correct parenthetical expressions of smaller size (perhaps even empty). C₂Cn-i for i=1 1. Hint: Any correctly matched set of n + 1 pairs of parenthesis
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