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 .

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