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)rightwards double arrow truth of P(n+1), n element of N.
On the basis of this, he could conclude that P(n) is true

  1. for no n    
  2. for all n greater or equal than 5    
  3. for all n    
  4. 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 n greater or equal than 5


    By induction the student proved:

    i) P(5) is true and
    ii) truth of P(n)rightwards double arrow truth of P(n+1), n element of N.
    rightwards double arrow space P left parenthesis 5 right parenthesis space space equals space P left parenthesis 6 right parenthesis space left parenthesis space F r o m space s t a t e m e n t space 2 right parenthesis
rightwards double arrow P left parenthesis 6 right parenthesis space equals space P left parenthesis 7 right parenthesis
rightwards double arrow P left parenthesis 7 right parenthesis space equals space P left parenthesis 8 right parenthesis
B u t comma space t h e space r e v e r s e space c a n n o t space b e space s a i d space f o r space t h e space s a m e space colon space i. e
w e space c a n n o t space p r o v e space b y space t h e s e space s t a t e m e n t s space t h a t space P left parenthesis n right parenthesis space equals space P left parenthesis n minus 1 right parenthesis
    Thus, by conclusion P(n) is true for all n greater or equal than 5

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