The equation which accurately solves for the integers is; 5n = 2(2n +3) +1.
It follows from the task content that the first number is represented by n.
On this note, it follows that the second, and third numbers are; (n+1) and (n+2) respectively.
On this note, since five times the first number is equal to one more than twice the sum of the second and third numbers; we have;
5n = 2(n+1+n+2) +1
5n = 2(2n+3)+1.
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