a) Prove that: Vne N, Σ" i2¹ = (n−1)2¹+¹ +2. i=0 b) Consider the sequence (Si)ie Z+ defined by s₁=1 and: Vne2..+∞, Sn=2Sn_1+n. Show that Sn 4sn_2+(2n+n)−2 and Sn=8Sn_3+(4n+2n+n)−(4×2+2). Likewise, express Sn in terms of Sn_4 and n, and then in terms of Sn_5 and n. c) Using a) and b), can you guess sn in terms of Sn-i, n and i? What expression do you get for Sn when i-n-1? Verify it by induction.

a) Prove that: Vne N, Σ" i2¹ = (n−1)2¹+¹ +2. i=0 b) Consider the sequence (Si)ie Z+ defined by s₁=1 and: Vne2..+∞, Sn=2Sn_1+n. Show that Sn 4sn_2+(2n+n)−2 and Sn=8Sn_3+(4n+2n+n)−(4×2+2). Likewise, express Sn in terms of Sn_4 and n, and then in terms of Sn_5 and n. c) Using a) and b), can you guess sn in terms of Sn-i, n and i? What expression do you get for Sn when i-n-1? Verify it by induction.

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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Question

4.2

a) Prove that: Vne N, Σ²i2¹ = (n−1)2¹+¹ +2.
i=0
b) Consider the sequence (Si)ie z+ defined by s₁=1 and: Vne 2..+∞, Sn=2Sn_1+n.
Show that Sn 4sn_2+(2n+n)-2 and sn=8Sn_3+(4n+2n+n)−(4×2+2).
Likewise, express sn in terms of Sn_4 and n, and then in terms of Sn_5 and n.
c) Using a) and b), can you guess sn in terms of Sn-i, n and i?
What expression do you get for Sn when i-n-1? Verify it by induction.

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