Question

Im so confused pls help

Answer 1

2,3

Explanation:

Let the first number be x

Hence, its next consecutive integer = x+1

`Reciprocal\ of\ the\ first\ integer\ = (1)/(x) \\Reciprocal\ of\ the\ second\ integer\ = (1)/(x+1) \\\\As\ we\ are\ given\ that\ their\ sums\ are\ (5)/(6) ,\\(1)/(x) + (1)/(x+1) = (5)/(6) \\\\Through\ butterfly\ method\ of\ resolving\ the\ denominators\,\\((x+1)+x)/(x(x+1)) = (5)/(6) \\(2x+1)/(x^2+x) =(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\\`

By Factorizing the expression on the LHS. we get it equal to :-

`5x^2-7x-6=0\\5x^2-10x+3x-6=0\\5x(x-2) + 3(x-2) = 0\\(5x+3)(x-2)=0`

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