This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.View Full Document
Unformatted text preview: n , (1 + 2 + ... + n ) 2 = 1 3 + 2 3 + ... + n 3 . 5. (a) Find a formula for n X k =1 1 2 k , and prove your answer by induction. (b) Find a formula for n X k =1 1 3 k , and prove your answer by induction. 6. Find a formula for the sum of the ﬁrst n fourth powers, and prove your answer by induction. Also explain how you were able to guess the right formula. 7. Give a proof or a counterexample for each. (a) Suppose that f : X → Y and g : X → Y . Then for all subsets B ⊆ Y , f ( g-1 [ B ]) = g ( f-1 [ B ]) . (b) Suppose that f : X → Y and g : X → Y . Then for all subsets A ⊆ X , f-1 [ g ( A )] = g-1 [ f ( A )] ....
View Full Document
This note was uploaded on 01/12/2010 for the course CALC 2b taught by Professor Doolittle during the Winter '06 term at Berkeley.
- Winter '06