Math 1B-­ Proof by Induction Project! Directions. Prove each statement using induction. (1) 1 + 2 + 3 + ... + n =
n(n + 1)
2
(2) 5 + 9 + 13 + ... + (4n + 1) = n(2n + 3) (3) 12 + 2 2 + 32 + ... + n 2 =
n(n + 1)(2n + 1)
6
(4) 2 3 + 4 3 + 6 3 + ... + (2n)3 = 2n 2 (n + 1)2 (5) 1⋅ 2 + 3 ⋅ 4 + 5 ⋅ 6 + ... + (2n − 1)(2n) =
n(n + 1)(4n − 1)
3
(6) 1
1
1
1
n(3n + 5)
+
+
+ ... +
=
1⋅ 3 2 ⋅ 4 3 ⋅ 5
n(n + 2) 4(n + 1)(n + 2)
(7) n ≤ 2 n −1 for all natural numbers n (8) Show 3 is a factor of n 3 + 2n for all natural numbers n (9) n 2 + 4 < (n + 1)2 for all natural numbers n ≥ 2 (10) n 3 > (n + 1)2 for all natural numbers n ≥ 3