{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# HW5_sol - Solutions to Homework 5 Section 4.1 4(a Plugging...

This preview shows pages 1–3. Sign up to view the full content.

Solutions to Homework 5 Section 4.1 4. (a) Plugging in n = 1 we have that P (1) is the statement 1 3 = [1 · (1 + 1) / 2] 2 . (b) Both sides of P (1) shown in part (a) equal 1. (c) The inductive hypothesis is the statement that 1 3 + 2 3 + · · · + k 3 = k ( k + 1) 2 2 . (d) For the inductive step, we want to show for each k 1 that P ( k ) implies P ( k +1) . In other words, we want to show that assuming the inductive hypothesis we can prove [1 3 + 2 3 + · · · + k 3 ] + ( k + 1) 3 = ( k + 1)( k + 2) 2 2 . (e) Replacing the quantity in brackets on the left-hand side of part (d) by what it equals by virtue of the inductive hypothesis, we have k ( k + 1) 2 2 + ( k + 1) 3 = ( k + 1) 2 k 2 4 + k + 1 = ( k + 1)( k + 2) 2 2 as desired. (f) We have completed both the basis step and the inductive step, so by the principle of mathematical induction, the statement is true for every positive integer n. 20. The basis step is n = 7 , and indeed 3 7 < 7! , since 2187 < 5040 . Assume the statement for k. Then 3 k +1 = 3 · 3 k < ( k + 1) · 3 k < ( k + 1) · k ! = ( k + 1)! , the statement for k + 1 . 1

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

View Full Document
38. The basis step is trivial, as usual: A 1 B 1 implies that S 1 j =1 A j S 1 j =1 B j because the union of one set is itself. Assume the inductive hypothesis that if A j B j for j = 1 , 2 , . . . , k , then S k j =1 A j S k j =1 B j . We want to show that if A j B j for j = 1 , 2 , . . . , k + 1 , then S k +1 j =1 A j S k +1 j =1 B j . To show that one set is a subset of another we show that an arbitrary element of the first set must be an element of the second set. So let x S k +1 j =1 A j = S k j =1 A j · A k +1 . Either x S k j =1 A j or x A k +1 . In the first case we know by the inductive hypothesis that x S k j =1 B j , in the second case, we know from the given fact that A k +1 B
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern