Every odd number can be written in a form 2k + 1, where k is integer thus for the five consecutive numbers we can write:

2k - 3, 2k - 1, 2k + 1, 3k + 3, 2k + 5, where k is integer

We need to find k so that the sum of the above numbers is 65 that is

(2k - 3) + (2k - 1) + (2k + 1) + (3k + 3) + (2k + 5) = 65

2k - 3 + 2k - 1 + 2k + 1 + 3k + 3 + 2k + 5 = 65

10k + 5 = 65

10k = 65 - 5

10k = 60

k = 60/10

k = 6

The first odd number will be 2k - 3 = 2*6 - 3 = 9

Indeed 9+11+13+15+17 = 65

The five consecutive odd numbers are 9, 11, 13, 15, 17.