Prove that the centre of the smallest circle passing through origin and whose centre lies on y = x + 1 is
Prove by induction that for n N, n2 + n is an even integer (n 1)
n = 1, n2 + n = 2 is an even integer
Let for n = k, k2 + k is even
Now for n = k + 1.
(k + 1)2 + (k + 1) - (k2 + k)
= k2 + 2k + 1 + k + 1 - k2 - k = 2k + 2
which is even integer, also k2 + k is integer
Hence (k + 1)2 + (k + 1) is also an even integer.
Hence n2 + n is even integer for all n N.