Update Fri Nov 8 00:54:05 EST 2024

Math2001/Homework/hw8.lean

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+/- Copyright (c) Heather Macbeth, 2024.  All rights reserved. -/
+import Mathlib.Tactic.GCongr
+import Library.Basic
+import AutograderLib
+
+macro_rules | `(tactic| gcongr_discharger) => `(tactic| numbers)
+math2001_init
+
+namespace Nat
+
+/-! # Homework 8
+
+Don't forget to compare with the text version,
+https://github.com/hrmacbeth/math2001/wiki/Homework-8,
+for clearer statements and any special instructions. -/
+
+
+def B : ℕ → ℚ
+  | 0 => 0
+  | n + 1 => B n + (n + 1 : ℚ) ^ 2
+
+@[autograded 4]
+theorem problem1 (n : ℕ) : B n = n * (n + 1) * (2 * n + 1) / 6 := by
+  sorry
+
+
+def S : ℕ → ℚ
+  | 0 => 1
+  | n + 1 => S n + 1 / 2 ^ (n + 1)
+
+@[autograded 4]
+theorem problem2 (n : ℕ) : S n = 2 - 1 / 2 ^ n := by
+  sorry
+
+
+def a : ℕ → ℤ
+  | 0 => 4
+  | n + 1 => 3 * a n - 5
+
+@[autograded 4]
+theorem problem3 : forall_sufficiently_large (n : ℕ), a n ≥ 10 * 2 ^ n := by
+  sorry
+
+
+def c : ℕ → ℤ
+  | 0 => 3
+  | 1 => 2
+  | n + 2 => 4 * c n
+
+@[autograded 4]
+theorem problem4 (n : ℕ) : c n = 2 * 2 ^ n + (-2) ^ n := by
+  sorry
+
+
+def q : ℕ → ℤ
+  | 0 => 1
+  | 1 => 2
+  | n + 2 => 2 * q (n + 1) - q n + 6 * n + 6
+
+@[autograded 4]
+theorem problem5 (n : ℕ) : q n = (n:ℤ) ^ 3 + 1 := by
+  sorry
+
+
+@[autograded 5]
+theorem problem6 (n : ℕ) (hn : 0 < n) : ∃ a x, Odd x ∧ n = 2 ^ a * x := by
+  sorry