( 1/2)^x+(2/3)^x+(5/6)^x= 1

When x=3, both sides are equal.

Let f (x) = (1/2) x+(2/3) x+(5/6) x.

Because 1/2, 2/3 and 5/6 are all greater than 0 and less than 1.

So (1/2) x, (2/3) x and (5/6) x are all decreasing functions.

So f(x) is a decreasing function.

And f(3)= 1.

So x>3, f (x) <; 1

x & lt3，f(x)> 1

So this equation only holds when x=3.

So x=3.