For, n = 1
P (n) = \( \dfrac{1}{1.2} = \dfrac{1}{1+1}\)
\( \dfrac{1}{2}= \dfrac{1}{2}\)
P (n) is true for n = 1
Let’s check for P (n) is true for n = k,
\(\dfrac{ 1}{1.2} + \dfrac{1}{2.3} +\dfrac{ 1}{3.4 }+ … + \dfrac{1}{k(k+1)} + \dfrac{k}{(k+1) (k+2)} = \dfrac{(k+1)}{(k+2)}\)
= \(\dfrac{ \dfrac{1}{(k+1)}}{(k+2)} + \dfrac{k}{(k+1)}\)
=\(\dfrac{ (k+1) }{ (k+2)}\)
P (n) is true for n = k + 1
Hence, P (n) is true for all n ∈ N.
Answered by Aaryan | 1 year agoGiven an example of a statement P (n) such that it is true for all n ϵ 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.