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Unformatted text preview: HOMEWORK 5 Question 1. Prove by induction that for all n ∈ N,
n i3 =
i=1 n(n + 1) 2 2 . Question 2. Consider the sequence deﬁned recursively by a1 = 2 an+1 = 7an + 9n + 5n Prove by induction that for all n ≥ 1, an = 9n − 5n . 2 Question 3. Consider the sequence deﬁned recursively by a1 = 1 a2 = 3 an+2 = 3an+1 − 2an Prove by induction that for all n ≥ 1, an = 2n − 1. Question 4. Consider the following 4 × 4 square grid from which one square has been removed: 1 2 HOMEWORK 5 Then it is easily checked that it can be covered without overlaps using Lshaped tiles of the following form: Prove that for any n ≥ 1, a 2n × 2n square grid with any one square removed can be covered without overlaps using such Lshaped tiles. ...
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 Spring '08
 ctw
 Math, square grid, Lshaped tiles

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