Q5 Use induction to show that for all positive integers n(a) 1³ + 2³ +33 + ... + n³ = (n(n + 1)/2)².(b) 1.1!+2.2!+ ... +n·n! = (n + 1)! - 1(c) if n > 6, then 3n

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Use induction to show that for all positive integers n (a) 13+23+33 + ... + n³ = (n(n+1)/2)². (b) 1. 1!+2.2!+ ... +n·n! = (n + 1)! - 1 (c) if n > 6, then 3n

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

Use induction to show that for all positive integers n
(a) 1³ + 2³ +33 + ... + n³ = (n(n + 1)/2)².
(b) 1.1!+2.2!+ ... +n·n! = (n + 1)! - 1
(c) if n > 6, then 3n <n!

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\Question No.2 Find f(1), f(2), S (3), and f(4) if s(n) is defined recur- sively by f(0) = 1 and for n 0, 1, 2, ... a) S(n + 1) = f(n) + 2. b) f(n + 1) -3/ (п). c) f(n + 1) = 2/W). d) S(n + 1) = (n)2 + S(n) + 1. Find f(1), f(2), f (3), f (4), and f (5) if f (n) is defined recursively by f (0) = 3 and for n =0, 1, 2, ... a) f(n + 1) = -2 (n). b) f (n + 1) = 3/ (n) +7. c) f(n + 1) = / (n)2 - 25 (n) – 2. d) f(n + 1) = 3/(m)/3. %3D Find f (2), f(3), f(4), and f(5) if f is defined recur- sively by f(0) =-1, S(1) = 2, and for n= 1, 2, ... a) f(n + 1) = S(n) + 35 (n – 1). b) f(n + 1) = f (n)?f (n – 1). c) f(n + 1) = 3/ (n)2 – 4f (n - 1)2. d) f (n + 1) = (n - 1)/S(n). Find f(2), S(3), (4), and f(5) if f is defined recur- sively by f(0) = (1) 1 and for n= 1, 2, ... a) S(n + 1) = f (n) - S(n – 1). b) f (n + 1) = f (n)/ (n - 1). c) f(n + 1) = / (n)2 + S(n – 1). d) f(n + 1) = r (n)/S(n - 1).