Homework2.pdf

# Homework2.pdf - Seul-Ki Kim ICS 6D Homework 2 1 Exercise...

• Homework Help
• 5

This preview shows 1 out of 3 pages.

Seul-Ki Kim ICS 6D January 23, 2019 Homework 2 1. Exercise 8.4.1 : Components of an inductive proof. (a-f) Define P(n) to be the assertion that: j ୨ୀଵ = ୬(୬ାଵ)(ଶ୬ା ) -------------------------------------------------------------------------------------------- (a) Verify that P(3) is true. 1 2 + 2 2 + 3 2 ? ଷ(ଷାଵ)(ଶ(ଷ)ାଵ) 1 + 4 + 9 ? ଷ(ସ)(଻) 14 = 14 (b) Express P(k). j ୨ୀଵ = ୩(୩ାଵ)(ଶ୩ାଵ) (c) Express P(k+1). j ୩ାଵ ୨ୀଵ = (୩ାଵ)(୩ାଵାଵ)(ଶ(୩ାଵ)ାଵ) = (୩ାଵ)(୩ାଶ)(ଶ୩ାଷ) (d) In an inductive proof that for every positive integer n, j ୨ୀଵ = ୬(୬ାଵ)(ଶ୬ାଵ) What must be proven in the base case? For the base case, it has to be proven that formula is true for n = 1. When n = 1, the left side of the equation is j ୨ୀଵ = 1 . When n = 1, the right side of the equation is ଵ(ଵାଵ)(ଶାଵ) = 1 Therefore, j ୨ୀଵ = ଵ(ଵାଵ)(ଶାଵ) . (e) In an inductive proof that for every positive integer n, j ୨ୀଵ = ୬(୬ାଵ)(ଶ୬ାଵ) What must be proven in the inductive step? Suppose that for positive integer k, j ୨ୀଵ = ୩(୩ାଵ)(ଶ୩ାଵ) , then we will show that j = (୩ାଵ)(୩ାଶ)(ଶ୩ାଷ) ୩ାଵ ୨ୀଵ Starting with the left side of the equation to be proven:

Subscribe to view the full document.

j = ∑ j + (k + 1) ୨ୀଵ ୩ାଵ ୨ୀଵ by separating out the last term = ୩(୩ାଵ)(ଶ୩ାଵ) + (k + 1) by the inductive hypothesis = ୩(୩ାଵ)(ଶ୩ାଵ) + ଺(୩ାଵ) = (୩ାଵ)[୩(ଶ୩ାଵ)ା଺(୩ାଵ)] = (୩ାଵ)[ଶ୩ ା଻୩ା଺] = (୩ାଵ)(୩ାଶ)(ଶ୩ାଷ) by algebra Therefore, j = (୩ାଵ)(୩ାଶ)(ଶ୩ାଷ) ୩ାଵ ୨ୀଵ (f)
You've reached the end of this preview.

{[ 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