Have a query? Ask a Tutor!!

Access to best educators and exclusive paper discussions

Can’t find what you’re looking for?

Our tutors are from top schools around the world, and they are waiting for you 24/7, anytime, anywhere.

Homework- One to One Solutions
Understand concepts to solve better
Get your doubts solved instantly

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

Related Questions to study

If $P space left parenthesis n right parenthesis$ is a statement $left parenthesis n element of N right parenthesis$ such that if $P space left parenthesis k right parenthesis$ is true, $P space left parenthesis k plus 1 right parenthesis$ is true for $k element of N$ , then $P space left parenthesis n right parenthesis$ is true.

If $P space left parenthesis n right parenthesis$ is a statement $left parenthesis n element of N right parenthesis$ such that if $P space left parenthesis k right parenthesis$ is true, $P space left parenthesis k plus 1 right parenthesis$ is true for $k element of N$ , then $P space left parenthesis n right parenthesis$ is true.

The resultant of two vectors is maximum when they

The resultant of two vectors is maximum when they

$straight C with rightwards arrow on top equals straight A with rightwards arrow on top plus straight B with rightwards arrow on top$ and A=B=C Find the angle between $A with not stretchy rightwards arrow on top$ and $Error converting from MathML to accessible text.$

$straight C with rightwards arrow on top equals straight A with rightwards arrow on top plus straight B with rightwards arrow on top$ and A=B=C Find the angle between $A with not stretchy rightwards arrow on top$ and $Error converting from MathML to accessible text.$

parallel

A and $straight B with rightwards arrow on top$ are nonzero vectors. Which from the followings is true?

A and $straight B with rightwards arrow on top$ are nonzero vectors. Which from the followings is true?

$straight A with rightwards arrow on top equals$ 1+2j+2k, calculate the angle between $A with not stretchy rightwards arrow on top$ and Y axis

$straight A with rightwards arrow on top equals$ 1+2j+2k, calculate the angle between $A with not stretchy rightwards arrow on top$ and Y axis

$A with not stretchy rightwards arrow on top$ and $Error converting from MathML to accessible text.$ are in opposite direction so they are

$A with not stretchy rightwards arrow on top$ and $Error converting from MathML to accessible text.$ are in opposite direction so they are

parallel

$straight P with rightwards arrow on top$ = $Error converting from MathML to accessible text.$ is true, if

$straight P with rightwards arrow on top$ = $Error converting from MathML to accessible text.$ is true, if

$Error converting from MathML to accessible text.$ and $Error converting from MathML to accessible text.$ are equal vectors what from the followings is true

$Error converting from MathML to accessible text.$ and $Error converting from MathML to accessible text.$ are equal vectors what from the followings is true

Which from the following is a scalar ?

Which from the following is a scalar ?

parallel

The greatest positive integer which divide $left parenthesis n plus 2 right parenthesis left parenthesis n plus 3 right parenthesis left parenthesis n plus 4 right parenthesis left parenthesis n plus 5 right parenthesis left parenthesis n plus 6 right parenthesis text if end text n element of N$ is

The greatest positive integer which divide $left parenthesis n plus 2 right parenthesis left parenthesis n plus 3 right parenthesis left parenthesis n plus 4 right parenthesis left parenthesis n plus 5 right parenthesis left parenthesis n plus 6 right parenthesis text if end text n element of N$ is

The sets $S subscript 1 comma S subscript 2 comma S subscript 3 comma horizontal ellipsis horizontal ellipsis$ are given by $S subscript 1 equals open parentheses 2 over 1 close parentheses comma S subscript 2 equals open parentheses 3 over 2 comma 7 over 2 close parentheses comma S subscript 3 equals open parentheses 4 over 3 comma 9 over 3 comma 14 over 3 close parentheses S subscript 4 equals open parentheses 5 over 4 comma 11 over 4 comma 17 over 4 comma 23 over 4 close parentheses horizontal ellipsis horizontal ellipsis$ then the sum of numbers in the set $S subscript 20$ is :

The sets $S subscript 1 comma S subscript 2 comma S subscript 3 comma horizontal ellipsis horizontal ellipsis$ are given by $S subscript 1 equals open parentheses 2 over 1 close parentheses comma S subscript 2 equals open parentheses 3 over 2 comma 7 over 2 close parentheses comma S subscript 3 equals open parentheses 4 over 3 comma 9 over 3 comma 14 over 3 close parentheses S subscript 4 equals open parentheses 5 over 4 comma 11 over 4 comma 17 over 4 comma 23 over 4 close parentheses horizontal ellipsis horizontal ellipsis$ then the sum of numbers in the set $S subscript 20$ is :

If $S subscript 1 equals left curly bracket 1 right curly bracket comma S subscript 2 equals left curly bracket 2 comma 4 right curly bracket comma S subscript 3 equals left curly bracket 3 comma 6 comma 9 right curly bracket S subscript 4 equals left curly bracket 4 comma 8 comma 12 comma 16 right curly bracket comma horizontal ellipsis horizontal ellipsis$
then sum of the numbers in the set is

If $S subscript 1 equals left curly bracket 1 right curly bracket comma S subscript 2 equals left curly bracket 2 comma 4 right curly bracket comma S subscript 3 equals left curly bracket 3 comma 6 comma 9 right curly bracket S subscript 4 equals left curly bracket 4 comma 8 comma 12 comma 16 right curly bracket comma horizontal ellipsis horizontal ellipsis$
then sum of the numbers in the set is

parallel

If $n$ is even, $S subscript n equals 1 squared plus 2.2 squared plus 3 squared plus 2.4 squared plus 5 squared plus 2.6 squared horizontal ellipsis n$ terms $equals fraction numerator n left parenthesis n plus 1 right parenthesis squared over denominator 2 end fraction$ , if $straight n$ is odd then $S subscript n equals$

If $n$ is even, $S subscript n equals 1 squared plus 2.2 squared plus 3 squared plus 2.4 squared plus 5 squared plus 2.6 squared horizontal ellipsis n$ terms $equals fraction numerator n left parenthesis n plus 1 right parenthesis squared over denominator 2 end fraction$ , if $straight n$ is odd then $S subscript n equals$

Central Rice Research Institute (CRRI) is situated at

Central Rice Research Institute (CRRI) is situated at

In the above figure $Error converting from MathML to accessible text.$ and $Error converting from MathML to accessible text.$ are two vectors. What from followings is true

In the above figure $Error converting from MathML to accessible text.$ and $Error converting from MathML to accessible text.$ are two vectors. What from followings is true

parallel

card img

With Turito Academy.

card img

With Turito Foundation.

card img

Get an Expert Advice From Turito.

Turito Academy

card img

With Turito Academy.

Test Prep

card img

With Turito Foundation.