Find three consecutive whole numbers such that the sum of the squares of the numbers is equal to 869?

Question

Find three consecutive whole numbers such that the sum of their squares equals 869. What are these numbers?

Anonymous · Accepted Answer

Numbers are: 16, 17. and 18

Anonymous · Answer

Let the 3 consecutive #s be n-1 , n , n+1. Sum of squares = 3n^2 +2. Then3n^2 = 867, ie., n^2 = 289 = (17)^2, ie., n = 17 & #s are 16,17,18.

Anonymous · Answer

You can estimate it very quickly.
Divide the sum by 3:
869/3 = 289 2/3
That must be close to the square of the middle number. Square root of that is 17.0195… The numbers must be  16 17 18.
Work out he square to verify that.

Anonymous · Answer

Let the consecutive whole number be &lsquo;n&rsquo;, &lsquo;n+1&rsquo;, &lsquo;n+2&rsquo; .Their squares are n^2 , (n+1)^2 , (n+2)^2 Their sum is n^2 + (n^2 + 2n + 1) + ( n^2 + 4n + 4) = 869 3n^2 + 6n + 5 = 869 3n^2 + 6n = 864 n^2 + 2n = 288(n + 1)^2 &ndash; (1)^2 = 288(n + 1)^2  = 289n + 1 = sqrt(289) = 17 Hence n = 16 & n + 2 = 18 Hence  the three consecutive numbers are 16,17,& 18.

Ms. Sarai Dickens V · Answer

x² + (x + 1)² + (x + 2)² = 869
x² + x² + 2x + 1 + x² + 4x + 4 = 869
3x² + 6x – 864 = 0
x² + 2x – 288 = 0
[ x + 18 ] [ x – 16 ] = 0
Accept x = 16
Integers are 16 , 17 and 18

Anonymous · Answer

Let  n – 1,   n,    and n + 1 be the consecutive numbers
Then
    (n – 1)^2  +  n^2  +  (n + 1)^2  = 869 
    n^2     –  2n  +  1  + n^2  +  n^2  +  2n  +  1  =  869
     3n^2    + 2   =  869
              3n^2    =  867
                  n^2    =  287
                    n    =   √287
                           =    17
   Numbers are:   16,  17.  and  18

Anonymous · Answer

(x &ndash; 1)^2 + x^2 + (x + 1)^2 = 869x^2 &ndash; 2x + 1 + x^2 + x^2 + 2x + 1 = 8693x^2 + 2 = 8693x^2 = 867x^2 = 289x = 17x &ndash; 1 = 16x + 1 = 18hence the numbers are: 16, 17, 18

Anonymous · Answer

Did you try 16, 17, and 18?
You would be adding three numbers to add up to 869.  So the average is 289.67.  The square root of 289.67 is 17.02.  So I would try numbers around 17.  The first set to try would be 16, 17, and 18.  This happens to work, so it’s the solution.

Helene Walsh

Find three consecutive whole numbers such that the sum of the squares of the numbers is equal to 869?

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