In three different experiments, the following results were obtained for the reaction
a -> products:
[a]o = 1.00m, t1/2=50 min ;
[a]o = 2.00m, t1/2 = 25 min;
[a]o = 0.50m, t1/2 = 100 min.

write the rate equation for this reaction, and indicate the value of k?

From the given observations,
You can see that as the concentration is doubled, half-life is halved.

That is,half-life is inversely proportional to concentration

As t( half-life) ~ 1/a^(n-1)

For this case n = 2,second order reaction.

R = k X a^n

Using the above formula you will get the rate and rate constant.

Yes you are correct use the scientific method