Please provide formal argument and detailed justification. Thank you!

Quadratic probing is one strategy we saw in class for handling collisions in hash tables. To find the next empty bucket in table T, we used h(k, i) = [h'(k) + cui + c2i2] mod m where h'(k) = k mod m and i = 0, 1, 2, ..., m 1. To simplify things, suppose that we drop the linear term from the probe sequence by setting c1 to 0. Our function for finding open buckets becomes [h'(k) + Czi] mod m for i=0,1,2, ..., m - 1. Show that this simplification is problematic when m is odd because the probe sequence will check at most (m +1)/2 buckets as i ranges from 0 to m - 1. Quadratic probing is one strategy we saw in class for handling collisions in hash tables. To find the next empty bucket in table T, we used h(k, i) = [h'(k) + cui + c2i2] mod m where h'(k) = k mod m and i = 0, 1, 2, ..., m 1. To simplify things, suppose that we drop the linear term from the probe sequence by setting c1 to 0. Our function for finding open buckets becomes [h'(k) + Czi] mod m for i=0,1,2, ..., m - 1. Show that this simplification is problematic when m is odd because the probe sequence will check at most (m +1)/2 buckets as i ranges from 0 to m - 1