If only 2 couples are existing today. Suppose each of these couple give birth to 4 children & the 8 children form 4 couples & each couple in turn give birth to 4 children, & those 16 form 8 couples & give birth to 4 children each & so on, can you tell exactly the number of persons existing after 10 generations, excluding the two initial couples?

Sure, let's break it down:

1. We start with 2 couples, which is 4 people. But we're excluding these from the final count.

2. Each couple has 4 children. So, in the first generation, we have 2 couples * 4 children = 8 people.

3. These 8 people form 4 couples, and each couple has 4 children. So, in the second generation, we have 4 couples * 4 children = 16 people.

4. This pattern continues for each generation. So, for each generation, the number of people is 2^n * 4, where n is the generation number.

5. We want to know the total number of people after 10 generations. So, we need to add up the number of people in each generation from 1 to 10.

6. This is a geometric series with a common ratio of 2. The sum of the first n terms of a geometric series is given by a * (r^n - 1) / (r - 1), where a is the first term, r is the common ratio, and n is the number of terms.

7. In this case, a = 8 (the number of people in the first generation), r = 2 (the common ratio), and n = 10 (the number of generations).

8. So, the total number of people after 10 generations is 8 * (2^10 - 1) / (2 - 1) = 8 * (1024 - 1) = 8 * 1023 = 8184 people.