tut8a-large

# tut8a-large - n-1 moves are required to assemble a puzzle...

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Problems to Week 9 Tutorial — MACM 101 1. Prove each of the following for all n 1 using the principal of mathematical induction. (a) 1 2 + 3 2 + 5 2 + ... + (2 n - 1) 2 = n (2 n - 1)(2 n + 1) 3 (b) n X i =1 1 i ( i + 1) = n n + 1 (c) n X i =1 i ( i !) = ( n + 1)! - 1 2. (Only for those familiar with complex numbers) Prove DeMoivre’s theorem (cos θ + i sin θ ) n = cos( ) + i sin( ) . 3. Prove that for all natural numbers n if n > 3 then 2 n < n !. 4. Let P ( n ) be the statement that a postage of n cents can be formed using just 3-cent stamps and 5-cent stamps. (a) Show that the statements P (8), P (9), and P (10) are true, completing the basis step of the proof. (b) What is inductive hypothesis of the proof? (c) What do you need to prove in the inductive step? (d) Complete the inductive step for k 10. 1

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5. A jigsaw puzzle is put together by successively joining pieces that ﬁt together into blocks. A move is made each time a piece is added to a block, or when two blocks are joined. Use strong induction to prove that no matter how the moves are carried out, exactly
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Unformatted text preview: n-1 moves are required to assemble a puzzle with n pieces. 6. Give a recursive deﬁnition of P m ( n ), the product of the integer m and the non-negative integer n . 7. A full binary tree is a graph deﬁned through the fol-lowing recursive deﬁnition. Basis step: A single vertex is a full binary tree. Inductive step: If T 1 and T 2 are disjoint full binary trees with roots r 1 , r 2 , respectively, the the graph formed by starting with a root r , and adding an edge from r to each of the vertices r 1 ,r 2 is also a full binary tree. Draw all full binary trees that can be obtain by ap-plying the inductive step at most 3 times (full binary trees of level 3). Use structural induction to show that n ( T ) ≥ 2 h ( T )+ 1, where n ( T ) denotes the number of vertices in T , and h ( T ) denotes the height of T . 2...
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tut8a-large - n-1 moves are required to assemble a puzzle...

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