# hw1-sol - CSC236H Introduction to the Theory of Computatoin...

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

CSC236H: Introduction to the Theory of Computatoin Homework 1 Solutions 1. Use induction to prove that the following equation holds for all positive integers n : n X k =1 1 k ( k + 1) = n n + 1 . Solution. We prove this by induction on n . Base case: Note that for n = 1 both sides of the equation equal 1 / 2. Inductive hypothesis: Let m N and suppose the statement is true for n = m - 1, i.e. suppose that m - 1 X k =1 1 k ( k + 1) = m - 1 m . Induction step: Using the inductive hypothesis we have m X k =1 1 k ( k + 1) = m - 1 X k =1 1 k ( k + 1) + 1 m ( m + 1) = m - 1 m + 1 m ( m + 1) = ( m - 1)( m + 1) + 1 m ( m + 1) = m m + 1 . Therefore the statement is also true for n = m . 2. Use induction to prove 3 n < n ! for all n > 6 with n N . Solution. Base case: The claim is made for n > 6. Therefore the base case we need to verify is n = 7. Since 3 7 = 2187 and 7! = 5040 the statement holds for the base case. Inductive hypothesis: Let

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 2

hw1-sol - CSC236H Introduction to the Theory of Computatoin...

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

View Full Document
Ask a homework question - tutors are online