Introduction:
We are given the expression 2.6.10.14... to n factors and we need to prove that it is equal to 2n! / n!(n-1)(n-2)...(2)(1).

Proof:
We will prove the given expression using mathematical induction.

Base Case:
Let's consider n = 1.
When n = 1, the expression 2.6.10.14... to n factors becomes 2.
On the other hand, 2n! / n!(n-1)(n-2)...(2)(1) becomes 2(1)! / 1!(1-1) = 2.
Hence, the expression is true for n = 1.

Inductive Hypothesis:
Assume that the expression is true for some positive integer k, i.e., 2.6.10.14... to k factors = 2k! / k!(k-1)(k-2)...(2)(1).

Inductive Step:
We need to prove that the expression holds for k+1, i.e., 2.6.10.14... to (k+1) factors = 2(k+1)! / (k+1)!(k)(k-1)...(2)(1).

Using the inductive hypothesis, we can write 2.6.10.14... to k factors as 2k! / k!(k-1)(k-2)...(2)(1).

Multiplying both sides of the equation by (k+1)(k)(k-1)...(2)(1), we get:
2.6.10.14... to k factors * (k+1)(k)(k-1)...(2)(1) = (2k! / k!(k-1)(k-2)...(2)(1)) * (k+1)(k)(k-1)...(2)(1).

Simplifying the equation, we have:
2.6.10.14... to k factors * (k+1)(k)(k-1)...(2)(1) = 2k! * (k+1).
Cancelling out the common terms, we get:
2.6.10.14... to k factors * (k+1)(k)(k-1)...(2)(1) = (k+1)(2k!).

Therefore, 2.6.10.14... to k factors * (k+1)(k)(k-1)...(2)(1) = (k+1)!

Hence, we have proved that the expression 2.6.10.14... to n factors = 2n! / n!(n-1)(n-2)...(2)(1) holds for all positive integers n by mathematical induction.