Is the following result consistent? Explain
$\displaystyle \sum{x^2}=64, \sum{x}=25, n=6$
Since it includes the information that "n= 6", I would interpret this as summing over 6 terms- and assume that the sums run from 1 to 6:
$\displaystyle \sum_{i= 0}^6 x_i^2= 64$ and $\displaystyle \sum_{i= 1}^6 x_i= 25$
So the question is "do there exist 6 numbers, that sum to 25, whose squares sum to 64?"
That is 6 numbers determined by only two equations so it seems to me there ought to be many ways to do that. For example, if we choose to take $\displaystyle x_1= 1$, $\displaystyle x_2= 2$, $\displaystyle x_3= 3$ and $\displaystyle x_4= 4$ the equations reduce to $\displaystyle 1+ 2+ 3+ 4+ x_5+ x_6= 25$ or $\displaystyle x_5+ x_6= 15$ and $\displaystyle 1+ 4+ 9+ 16+ x_5^2+ x_6^2= 64$ or $\displaystyle x_5^2+ x_6^2= 34$. Since $\displaystyle x_5= 16- x_6$ so that $\displaystyle x^5^2+ x_6^2= x_6^2- 32x_6+ 64+ x_6^2= 34$. $\displaystyle x_6^2- 16x_6= -30$.
Was there a requirement that the numbers be integers?