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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 :

for $n > 1$
for all $n \in N$
for $n > 2$
for no $n \in N$

hint 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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