Question

Let P(n) be a statement and $P\left( n \right) = P\left( {n + 1} \right)\forall n \in N$, then P(n) is true for what values of n?
A For all n
B For all $n > 1$
C For all $n > m$, m being a fixed positive integer
D Nothing can be said

Answer 1

Here, in the question, $n \in N$ which means n belongs to N. Here N is an indication of natural numbers. So, n belongs to set N which means n is a natural number and as per the given data P(n) consists of the values which are true for all the natural numbers. Natural numbers are a part of the number system which includes all the positive integers from 1 till infinity and are also used for counting purposes and it does not include zero.

Complete step by step solution:
Let us write the given data as
$P\left( n \right) = P\left( {n + 1} \right)\forall n \in N$
To find out the values of n, which satisfies with P(n) of the given statement, hence substitute n-1 in place of n i.e.,
$P\left( n \right) = P\left( {n + 1} \right)$
$P\left( {n - 1} \right) = P\left( {n - 1 + 1} \right)$
$P\left( n \right) - 1 = P\left( n \right)$
Thus, if P(n) is true for $n \in N$, then it is true for n-1 and n+1.
Therefore, P(n) is true for $\forall n \in N$ and hence, option A is the right answer.

Additional information:
Properties of natural numbers are:

1. Closure property
2. Commutative property
3. Associative property
4. Distributive property

Note:
Natural numbers include all the whole numbers excluding the number 0. In other words, all-natural numbers are whole numbers, but all whole numbers are not natural numbers. Zero does not have a positive or negative value. Since all the natural numbers are positive integers, hence we cannot say zero is a natural number. Although zero is called a whole number.