Probability that none of the events will occur:
To find the probability that none of the events will occur, we need to find the probability of the complement of the union of all the events.

Complement of the union of events:
The complement of the union of events is the event that none of the events occur. Let's denote this event as A'.

Using the complement rule:
According to the complement rule of probability, the probability of an event A' is equal to 1 minus the probability of the event A.

Probability of the union of events:
To find the probability of the union of all the events, we can use the principle of inclusion-exclusion.

Inclusion-Exclusion Principle:
The principle of inclusion-exclusion states that the probability of the union of two events A and B is given by:
P(A U B) = P(A) + P(B) - P(A ∩ B)

Applying the inclusion-exclusion principle:
Using the inclusion-exclusion principle, we can find the probability of the union of n events A1, A2, ..., An as follows:
P(A1 U A2 U ... U An) = P(A1) + P(A2) + ... + P(An) - P(A1 ∩ A2) - P(A1 ∩ A3) - ... - P(An-1 ∩ An) + P(A1 ∩ A2 ∩ A3) + ... + (-1)^(n-1) * P(A1 ∩ A2 ∩ ... ∩ An)

Probability of each individual event:
Given that P(Ai) = 1/(i+1), we can substitute these probabilities into the formula for the probability of the union of events.

Calculating the probability:
P(A1 U A2 U ... U An) = 1/2 + 1/3 + ... + 1/(n+1) - 1/2∩3 - 1/2∩4 - ... - 1/(n-1)∩n + 1/2∩3∩4 + ... + (-1)^(n-1) * 1/2∩3∩...∩(n+1)

Simplifying the expression:
After simplifying the expression, we can observe that all the terms in the sum will cancel out, leaving us with only 1/(n+1).

Probability that none of the events will occur:
Finally, we can use the complement rule to find the probability of the event A', which is the event that none of the events A1, A2, ..., An occur.
P(A') = 1 - P(A1 U A2 U ... U An)
P(A') = 1 - 1/(n+1)
P(A') = n/(n+1)

Therefore, the probability that none of the events will occur is n/(n+1), which can be simplified to 1/(n+1). Hence, the correct answer is option 'C'.