Maths-
General
Easy
Question
A student was asked to prove a statement by induction. He proved
i) P(5) is true and
ii) truth of P(n)
truth of P(n+1), 
.
On the basis of this, he could conclude that P(n) is true
- for no n
- for all
- for all n
- nothing can be said
Hint:
Consider a statement P(n), where n is a natural number. Then to determine the validity of P(n) for every n, use the following principle:
Step 1: Check whether the given statement is true for n = 1.
Step 2: Assume that given statement P(n) is also true for n = k, where k is any positive integer.
Step 3: Prove that the result is true for P(k+1) for any positive integer k.
If the above-mentioned conditions are satisfied, then it can be concluded that P(n) is true for all n natural numbers.
The correct answer is: for all
By induction the student proved:
i) P(5) is true and
ii) truth of P(n) truth of P(n+1), .
Thus, by conclusion P(n) is true for all .
ii) truth of P(n)
truth of P(n+1), .Thus, by conclusion P(n) is true for all
. 
