Consider the following figure

The letters in the little triangles stand for the surface areas of the respective triangles. Identical letters denote identical areas of triangles having the same base lengths and the same heights.

We are talking about medians, so we have further equations:

$$2a+b=2c+b.$$

From here, it follows that $c=a$.

Then, we have

$$2a+c=2b+c.$$

As a result $b=a$.

So, $$b=a=c.$$

And this is what we were to prove.

You have the right approach. The only missing part is an O(1) way to decide which side you are discarding.

First, note that the numbers in your problem must be ordered, otherwise you can't do better than O(n). There also needs to be at least three numbers, otherwise you wouldn't figure out the "step".

With this understanding in place, you can determine the "step" in O(1) time by examining the initial three terms, and see what's the difference between the consecutive ones. Two outcomes are possible:

• Both differences are the same, and
• One difference is twice as big as the other.

Case 2 hands you a solution by luck, so we will consider only the first case from now on. With the step in hand, you can determine if the range has a gap in it by subtracting the endpoints, and comparing the result to the number of gaps times the step. If you arrive at the same result, the range does not have a missing term, and can be discarded. When both halves can be discarded, the gap is between them.