# h1 - 1 Homework I 1.9 Find a formula for 1 is correct n j...

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

1 Homework I: June 19, 2009 1 . 9 . Find a formula for 1 + n j =1 j ! j ; use induction to prove that your formula is correct. A list of the sums for n = 1 , 2 , 3 , 4 , 5 is 2 , 6 , 24 , 120 , 720. These are factorials; better, they are 2! , 3! , 4! , 5! , 6!. If we write f ( n ) = 1 + n j =1 j ! j , then our guess is f ( n ) = ( n + 1)!. We have already checked the base step n = 1, for f (1) = 2 = 2!. For the inductive step, we must prove f ( n + 1) = 1 + n +1 X j =1 j ! j = ( n + 2)! . Rewrite the middle expression: h n X j =1 j ! j i + ( n + 1)!( n + 1) . The inductive hypothesis says that the bracketed term is ( n + 1)!, and so f ( n + 1) = ( n + 1)! + ( n + 1)!( n + 1) = ( n + 1)!( n + 2) = ( n + 2)! . By induction, f ( n ) = ( n + 1)! for every n 1. 1 . 47 . Given integers a and b (possibly negative) with a 6 = 0, prove that there exist unique integers q and r with b = aq + r and 0 r < | a | . We have already proved this in class when

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

h1 - 1 Homework I 1.9 Find a formula for 1 is correct n j...

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

View Full Document
Ask a homework question - tutors are online