every odd integer can be written as 2k + 1, where k is integer

the 3 consecutive odd integers can be written as:

2k - 1, 2k + 1, 2k + 3

the largest is 2k + 3

the smallest is 2k - 1

5 times the largest is 114 more than twice the smallest

5(2k + 3) = 114 + 2(2k - 1)

by solving we find

k = 97/6

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since the solution we got k = 97/6 is not integer follows that there are no such consecutive odd numbers that is the problem has no solution.