Midterm - SOLUTIONS FOR MIDTERM 1 (10 points) : Prove that,...

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

View Full Document Right Arrow Icon

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

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

Unformatted text preview: SOLUTIONS FOR MIDTERM 1 (10 points) : Prove that, for any positive integer n , n +1 summationdisplay k =1 k 3 = ( n + 1) 2 ( n + 2) 2 4 . Define Proposition P m : ∑ m k =1 k 3 = m 2 ( m + 1) 2 / 4, and show that P m holds for any m ∈ N (the equality in the statement of the problem is P n +1 ). We use induction. The base is easy to establish for m = 1. Indeed, P 1 states that 1 3 = 1 2 (1 + 1) 2 / 4, which is true. The induction step consists of proving that P n implies P n +1 . To this end, assume ∑ n k =1 k 3 = n 2 ( n + 1) 2 / 4, and show that ∑ n +1 k =1 k 3 = ( n + 1) 2 ( n + 2) 2 / 4. We have: n +1 summationdisplay k =1 k 3 = n summationdisplay k =1 k 3 + ( n + 1) 3 = n 2 ( n + 1) 2 4 + ( n + 1) 3 = ( n + 1) 2 4 parenleftBig n 2 + 4( n + 1) parenrightBig = ( n + 1) 2 4 parenleftBig n 2 + 4 n + 4 parenrightBig = ( n + 1) 2 ( n + 2) 2 4 . 2 (10 points) : Suppose S is a subset of (0 , ∞ ), bounded above. Let S- 1 = { s- 1 | s ∈ S } . Prove that S- 1 is bounded below, and inf...
View Full Document

This note was uploaded on 05/08/2010 for the course MATH 140a taught by Professor Staff during the Winter '08 term at UC Irvine.

Page1 / 2

Midterm - SOLUTIONS FOR MIDTERM 1 (10 points) : Prove that,...

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