Give an example of a statement \(P(n)\) which is true for all \(n \ge 4\) but \(P(1)\), \(P(2)\) and \(P(3)\) are not true. Justify your answer.
\(P(n): 2n < 2^n\)
Give an example of a statement \(P(n)\) which is true for all \(n\). Justify your answer.
\(P(n): 1 + 2 + 3 + \dots + n = \dfrac{n(n+1)}{2}\)
\(4^n - 1\) is divisible by 3, for each natural number \(n\).
\(2^{3n} - 1\) is divisible by 7, for all natural numbers \(n\).
\(n^3 - 7n + 3\) is divisible by 3, for all natural numbers \(n\).
\(3^{2n} - 1\) is divisible by 8, for all natural numbers \(n\).
For any natural number \(n\), \(7^n - 2^n\) is divisible by 5.
For any natural number \(n\), \(x^n - y^n\) is divisible by \(x - y\), where \(x\) and \(y\) are integers with \(x \ne y\).
\(n^3 - n\) is divisible by 6, for each natural number \(n \ge 2\).
\(n(n^2 + 5)\) is divisible by 6, for each natural number \(n\).
\(n^2 < 2^n\) for all natural numbers \(n \ge 5\).
\(2n < (n + 2)!\) for all natural numbers \(n\).
\(\sqrt{n} < \dfrac{1}{\sqrt{1}} + \dfrac{1}{\sqrt{2}} + \dots + \dfrac{1}{\sqrt{n}}\), for all natural numbers \(n \ge 2\).
\(2 + 4 + 6 + \dots + 2n = n^2 + n\) for all natural numbers \(n\).
\(1 + 2 + 2^2 + \dots + 2^n = 2^{n+1} - 1\) for all natural numbers \(n\).
\(1 + 5 + 9 + \dots + (4n - 3) = n(2n - 1)\) for all natural numbers \(n\).