Ans-9 - Answers, Assignment #9 Exercises 1.1. 1. Let S be...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Answers, Assignment #9 Exercises 1.1. 1. Let S be the set of integers for which the statement is true. Since 1 = 1 · 2 2 =1 , 1 S. Assume that the statement is true for some positive integer n = k Then, for n = k +1, 1+2+ ··· + k +( k +1) = [1+2+ + k ]+( k +1)= k ( k +1) 2 k = k ( k + 1) + 2( k 2 = ( k + 1)( k +2) 2 Thus, k +1 S and the statement is true for all n N . 2. Let S be the set of integers for which the statement is true. Since 1 2 =1= 1 · 2 · 3 6 = 6 6 , 1 S. Assume that the statement is true for some positive integer n = k Then, for n = k 1 2 +2 2 + + k 2 k 2 =[ 1 2 2 + + k 2 k 2 = k ( k + 1)(2 k 6 k 2 = k ( k + 1)(2 k +1)+6( k 2 6 = ( k + 1)( k + 2)(2[ k +1]+1) 6 Thus, k S and the statement is true for all n N . 3. Let S be the set of integers for which the statement is true. Since 1 = 1 - r 1 - r , 1 S. Assume that the statement is true for some positive integer n = k Then, for n = k 1+ r + + r k + r k +1 1 + r + + r k ]+ r k +1 = 1 - r k +1 1 - r + r k +1 = 1 - r k +1 +(1 - r ) r k +1 1 - r = 1 - r k +2 1 - r Thus, k S and the statement is true for all n N . 4. Let S be the set of integers for which the statement is true.
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 01/06/2011 for the course MATH 3333 taught by Professor Staff during the Spring '08 term at University of Houston.

Page1 / 3

Ans-9 - Answers, Assignment #9 Exercises 1.1. 1. Let S be...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online