Show that k2+k+1 is not a square number for k€N

 

Solution,

For any natural number k,

K²< k²+k+1 < k²+2k+1

K²< k²+k+1 < (k+1)²             …………………..(1)say

Here it is known to us that no square number is found between two consecutive square number.

 

If possible,

Assume an integer r such that, k²+k+1= r²

From,(1)  

    K²< k²+k+1 < (k+1)² 

 = K²< r² < (k+1)²

  = K< r < k+1

Since no such integer lies between 2 consecutive integers.

So our assumption is wrong that,

 k²+k+1= r²

Hence k²+k+1 is not a square number for k € N