nasex2 - M 2003 NUMBERS AND SETS EXAMPLES SHEET 2 W. T. G....

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M 2003 NUMBERS AND SETS – EXAMPLES SHEET 2 W. T. G. 1. Prove by induction that the following two statements are true for every positive inte- ger n . (i) The number 2 n +2 + 3 2 n +1 is a multiple of 7. (ii) 1 3 + 3 3 + 5 3 + . . . + (2 n - 1) 3 = n 2 (2 n 2 - 1) . 2. Suppose that you have a 2 n × 2 n grid of squares (if n = 3 then you have a chessboard) and you remove one square. Prove that, wherever the removed square is, the remaining squares can be tiled with L-shaped tiles - that is, tiles consisting of three squares that form a 2 × 2 grid with one square removed. 3. By considering the equation (1 - 1) n = 0, give another proof that exactly half the subsets of { 1 , 2 , . . . , n } have even size. 4. Prove that ± k k ² + ± k + 1 k ² + . . . + ± n + k - 1 k ² = ± n + k k + 1 ² for any n > k . [Hint: for each set of size k + 1 consider its largest element.] 5. Prove that ± n 0 ² 2 + ± n 1 ² 2 + ± n 2 ² 2 + . . . + ± n n ² 2 = ± 2 n n ² . [Hint:
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This note was uploaded on 03/06/2012 for the course MATH 508 taught by Professor Staff during the Fall '10 term at UPenn.

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nasex2 - M 2003 NUMBERS AND SETS EXAMPLES SHEET 2 W. T. G....

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