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if n stands for a perfect square, what formula stands for the next perfect square?

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2 Answers

If n is a perfect square, then the square root of n is an integer. For example, if n = 36, then the square root of n = 6. The next perfect square then is:
 
(√n + 1)2
 
Staying with the example of n = 36, the square root of n = 6, and the expression yields:
 
(√36 + 1)2 = (6 + 1)2  = 72 = 49
            
n^2 = ∑n    (2ni + 1 )    / Square of any number n equals Sum of the Odd numbers from 1 to n
          i=0
                       i.e.  3^2 = 1 + 3 + 5 =9
   
     Next square number to n^2  equals n^2 +2n +1
 
 
                                 ie.  3 ^2 = 9      9 + 2(3) +1= 16 = 4^2 .