1. Use induction to prove Bernoulli’s inequality: If 1 + x > 0, then (1 + x) n ≥1 + n x for all n ∈ . 2. Prove that for all integers x ≥ 8, x can be written in the form 3m + 5n, where m and n are...

1. Use induction to prove Bernoulli’s inequality: If 1 + x > 0, then (1 + x)ⁿ
≥1 + n x for all n ∈
.

2. Prove that for all integers x ≥ 8, x can be written in the form 3m + 5n, where m and n are nonnegative integers

3. Consider the statement “For all integers x ≥ k, x can be written in the form 5m + 7n, where m and n are nonnegative integers.”

(a) Find the smallest value of k that makes the statement true.

(b) Prove the statement is true with k as in part (a).