State the first principle of mathematical induction.

Using principle of mathematical induction prove that

for all natural numbers n ≥ 2.

The distributive law from algebra states that for real numbers

c, a₁ and a₂, we have c(a₁ + a₂) = c a₁ + ca₂

Use this law and mathematical induction to prove that, for all

natural numbers, n ≥ 2, if c, a₁, a₂, …... a_n are any real numbers,

then c(a₁ + a₂ +…+ a_n) = c a₁ + c a₂ +…+ c a_n.

Write the set of value of n for which the statement P(n): 2n < n! is true.

State the second principle of mathematical induction.