Question -
Answer -
Let us consider P (n) = 1 + 2 + 3 + ….. + n = n (n +1)/2
For, n = 1
LHS of P (n) = 1
RHS of P (n) = 1 (1+1)/2 = 1
So, LHS = RHS
Since, P (n) is true for n = 1
Let us consider P (n) be the true for n = k, so
1 + 2 + 3 + …. + k = k (k+1)/2 … (i)
Now,
(1 + 2 + 3 + … + k) + (k + 1) = k (k+1)/2 + (k+1)
= (k + 1) (k/2 + 1)
= [(k + 1) (k + 2)] / 2
= [(k+1) [(k+1) + 1]] / 2
P (n) is true for n = k + 1
P (n) is true for all n ∈N
So, by the principle of Mathematical Induction
Hence, P (n) = 1 + 2 + 3 + ….. + n = n (n +1)/2 is true forall n ∈ N.