1. Use the principle of mathematical induction to prove that
(a) n2 ≤ 2n for all n ≥ 5
(b) 72n − 48n − 1 is divisible by 2304.
(c) (1 + x)n > 1 + nx.
2. Use the definition of a limit to prove that
n
=0
(a) lim
n→∞ n2 + 1
4n + 3
4
(b) lim
=
n→∞ 7n − 5
7
p
1
(c) lim
4n2 + n − 2n =
n→∞
4
4 − 2n
2
(d) lim
=−
n→∞ 3n + 2
3
3. Use the property of limits to simplify the following
4 − 2n − 3n2
(a) lim
n→∞
2n2 + n
!
√
3n2 − 5n + 4
(b) lim
n→∞
n−7
p
(c) lim
4n2 + n + 5 − 2n
n→∞
1