Answer:
2,3
Step-by-step explanation:
Let the first number be x
Hence, its next consecutive integer = x+1
[tex]Reciprocal\ of\ the\ first\ integer\ = \frac{1}{x} \\Reciprocal\ of\ the\ second\ integer\ = \frac{1}{x+1} \\\\As\ we\ are\ given\ that\ their\ sums\ are\ \frac{5}{6} ,\\\frac{1}{x} + \frac{1}{x+1} = \frac{5}{6} \\\\Through\ butterfly\ method\ of\ resolving\ the\ denominators\,\\\frac{(x+1)+x}{x(x+1)} = \frac{5}{6} \\\frac{2x+1}{x^2+x} =\frac{5}{6} \\6(2x+1)=5(x^2+x)\\12x+6=5x^2+5x\\\\Now, lets\ bring\ all\ the\ terms\ in\ the\ LHS\ to\ RHS.\\\\Hence,\\0=5x^2+5x-12x-6\\5x^2-7x-6=0\\[/tex]
By Factorizing the expression on the LHS. we get it equal to :-
[tex]5x^2-7x-6=0\\5x^2-10x+3x-6=0\\5x(x-2) + 3(x-2) = 0\\(5x+3)(x-2)=0[/tex]
Hence, we get two solutions for x, x=2, x= -3/5
As x is an integer, x=2 is the most suitable solution in this case.
Hence,
The numbers are 2,3
