EDB — 1XG

[1XG] Prove ¹ by induction the following assertions:

1. \(∑_{k=1}^ nk=\frac{n(n+1)} 2\);

2. \(∑_{k=1}^ nk^ 2=\frac{n(n+1)(2n+1)} 6\);

3. \(∑_{k=1}^ nk^ 3=\frac{n^ 2(n+1)^ 2} 4\);

4. \(∑_{k=1}^ n\frac{1}{4k^ 2-1}=\frac{n}{2n+1}\);

5. \(∑_{k=1}^ n\frac{k}{2^ k}=2-\frac{n+2}{2^ n}\);

6. \(n!≥ 2^{n-1}\);

7. If \(x{\gt}-1\) is a real number and \(n∈ ℕ\) then \((1+x)^ n≥ 1+nx\) (Bernoulli inequality).