Given an example of a statement P (n) such that it is true for all n ϵ N.
Let us consider
P (n) = 1 + 2 + 3 + – – – – – + n = \(\dfrac{ n(n+1)}{2}\)
So,
P (n) is true for all natural numbers.
Hence, P (n) is true for all n ∈ N.
a + (a + d) + (a + 2d) + … + (a + (n-1)d) = \( \dfrac{n}{2}\) [2a + (n-1)d]
72n + 23n – 3. 3n – 1 is divisible by 25 for all n ϵ N
n (n + 1) (n + 5) is a multiple of 3 for all n ϵ N.
(ab) n = an bn for all n ϵ N
32n + 2 – 8n – 9 is divisible by 8 for all n ϵ N