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(b) Show that k-1 (k+1) n n+1 for every n ≥ 1.

(b) Show that k-1 (k+1) n n+1 for every n ≥ 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 b Proof by induction

4. Proof by induction: (a) Prove that 3 (2n+1 +52) for every integer n ≥ 1. n (b) Show that k-1 k(k+1) = for every n ≥ 1. n+1 (c) For every n EN let Gn be a graph constructed according to the following procedure: • Go consists of a single vertex vo and no edges. . Gn is obtained from Gn-1 by adding a vertex Un, picking k € {0, 1, 2,..., n-1}, and adding an edge between Un and Uk. Show that Gn is a tree on n + 1 vertices for every n E N, no matter what number k we pick in cach step.
Transcribed Image Text:4. Proof by induction: (a) Prove that 3 (2n+1 +52) for every integer n ≥ 1. n (b) Show that k-1 k(k+1) = for every n ≥ 1. n+1 (c) For every n EN let Gn be a graph constructed according to the following procedure: • Go consists of a single vertex vo and no edges. . Gn is obtained from Gn-1 by adding a vertex Un, picking k € {0, 1, 2,..., n-1}, and adding an edge between Un and Uk. Show that Gn is a tree on n + 1 vertices for every n E N, no matter what number k we pick in cach step.
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