Question

prove that the value of the expression n^2-3n+1 /n-1 is not a prime number for all n is natural numbers N

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Answer 1

Given :   ( n² - 3n + 1) / (n - 1)

To Find :  prove that the value of expression is not a prime number for all n€N​

Solution:

( n² - 3n + 1) / (n - 1)

=  ( n² -2n - n + 1) / (n - 1)

=   ( n² -2n   + 1 - n) / (n - 1)

= ( (n - 1)²  - n) / (n - 1)

=  n - 1  -  n/(n - 1)

to be a prime number  n/(n - 1) should be integer

which is only possible  if  n = 2

for n = 2

Value s

2  -  1  -  2/1

= 2 - 1 - 2

= - 1    which is not a prime number

Hence for any other value of  n€N​     ( n² - 3n + 1) / (n - 1) is not even integer

Hence can not be a prime number.

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Answer 2

SOLUTION

TO PROVE

The below expressions is not a prime number for all n

$\displaystyle \sf{ \frac{ {n}^{2} - 3n + 1 }{n - 1} }$

EVALUATION

Here the given expression is

$\displaystyle \sf{ \frac{ {n}^{2} - 3n + 1 }{n - 1} }$

The above expression is undefined when n = 1

So the expression is not defined for all values of n

Moreover if we take n = 3 then we get

$\displaystyle \sf{ = \frac{ {3}^{2} - 3 \times 3 + 1 }{3 - 1} }$

$\displaystyle \sf{ = \frac{9- 9 + 1 }{2} }$

$\displaystyle \sf{ = \frac{ 1 }{2} }$

Which is not a prime

Hence

$\displaystyle \sf{ \frac{ {n}^{2} - 3n + 1 }{n - 1} }$

is not a prime number for all n

Hence proved

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