1.1 Prove 12 + 22 + · · ·+ n2 = 1
6 n(n + 1)(2n + 1) for all positive integers n.
1.2 Prove 3 + 11 + · · · + (8n − 5) = 4n2 − n for all positive integers n.
1.3 Prove 13 + 23 + · · · + n3 = (1 + 2 + · · · + n)2 for all positive integers n.
1.4 (a) Guess a formula for 1 + 3 + · · · + (2n − 1) by evaluating the sum
for n = 1, 2, 3, and 4. [For n = 1, the sum is simply 1.]
(b) Prove your formula using mathematical induction.
1.5 Prove 1 + 1
2 + 1
4 + · · · + 1
2n = 2 − 1
2n for all positive integers n.
1.6 Prove (11)n − 4n is divisible by 7 when n is a positive integer.
1.7 Prove 7n − 6n − 1 is divisible by 36 for all positive integers n.
1.8 The principle of mathematical induction can be extended as follows.
A list Pm, Pm+1, . . . of propositions is true provided (i) Pm is true,
(ii) Pn+1 is true whenever Pn is true and n ≥ m.
(a) Prove n2 > n + 1 for all integers n ≥ 2.
(b) Prove n! > n2 for all integers n ≥ 4. [Recall n! = n(n − 1) · · · 2 · 1;
for example, 5! = 5 · 4 · 3 · 2 · 1 = 120.]
1.9 (a) Decide for which integers the inequality 2n > n2 is true.
(b) Prove your claim in (a) by mathematical induction.
1.10 Prove (2n + 1) + (2n + 3) + (2n + 5) + · · · + (4n − 1) = 3n2 for all
positive integers n.
1.11 For each n ∈ N, let Pn denote the assertion “n2 + 5n + 1 is an even
integer.”
(a) Prove Pn+1 is true whenever Pn is true.
(b) For which n is Pn actually true? What is the moral of this
exercise?
