If a sample is mixed with more than one radioactive elements - for example Ra-226, Po - 210 and Cs -137, then how to estimate the resultant half life of the sample? Is there any general formula to calculate the resultant half life?
There are several things to consider. The amount of each nuclide in the mixture will determine the overall activity at any particular time.
The total activity of a mixture of 3 nuclides would be $$A = \lambda_1 N_1 + \lambda_2 N_2 + \lambda_3 N_3 $$.
Because the $N$s are changing at different rates, there will not be a single resultant halflife for the mixture. If you know the individual activities or populations of each, you can calculate how much time is needed for the total activity to become half, but that won't be a constant halflife in the sense we normally mean.
With multiple nuclide types, the radioactivity is no longer described by a simple exponential, and as a result no longer has a simple half-life. Consider a rock which has equal numbers of two isotopes A and B, A with half-life of 1000 years and B with 1 million years, each with the same emission energies. The immediate decay level of A will be 1000 times that of B, so the overall emission will be dominated by A, and will reach the halfway point in slightly more than 1000 years - in rough terms, 1001 years. However, in about 10,000 years the emissions from A will drop below those of B, and from then on B will dominate the activity of the sample, which will not conform to the expectations which would result from applying the 1001 year half-life to the standard model of exponential decay.