Maths-
General
Easy

Question

Let P space left parenthesis n right parenthesis colon n squared plus n plus 1 is an even integer. If P space left parenthesis k right parenthesis is assumed true rightwards double arrow P space left parenthesis k plus 1 right parenthesis is true. Therefore P left parenthesis n right parenthesis is true :

  1. for n greater than 1    
  2. for all n element of N    
  3. for n greater than 2    
  4. for no n element of N    

Hint:

Mathematical reasoning or the principle of mathematical reasoning is a part of mathematics where we determine the truth values of the given statements.

The correct answer is: for no n element of N


    Given :  P space left parenthesis n right parenthesis colon n squared plus n plus 1 is an even integer
    Let this be true for n = k
    P(k) equals space k squared space plus space k space plus 1 is an even integer
    This is also true for P(k +1)
    P(k +1) = open parentheses k space plus space 1 close parentheses squared space plus space left parenthesis k plus 1 right parenthesis space plus space 1
    P(k +1) = k squared space plus space 1 space plus 2 k space plus space left parenthesis k plus 1 right parenthesis space plus space 1

    P(k +1) = k squared space plus space k space plus space 1 space plus space 2 k space plus 2
    w e space k n o w space t h a t space space k squared space plus space k space plus space 1 space i s space a n space e v e n space i n t e g e r
    P left parenthesis k space plus 1 right parenthesis space space equals space e v e n space plus space 2 left parenthesis K space plus 1 right parenthesis
2 left parenthesis k space plus 1 right parenthesis space w i l l space a l s o space b e space e v e n
P left parenthesis k plus 1 right parenthesis space equals space e v e n space plus space e v e n space equals space e v e n space i n t e g e r
    P left parenthesis n right parenthesis space equals space n squared space plus space n space plus 1
P left parenthesis n right parenthesis space equals space n left parenthesis n plus 1 right parenthesis space plus space 1 comma
w h e r e space n left parenthesis n plus 1 right parenthesis space w i l l space a l w a y s space b e space a n space e v e n space i n t e g e r
T h u s space w h e n space w e space a d d space 1 space t o space e v e n space i n t e g e r space i t space b e c o m e s space o d d
P left parenthesis n right parenthesis space i s space o d d space i n t e g e r
T h u s comma space P left parenthesis n right parenthesis space i s space t r u e space f o r space n o space n space element of space N.

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