the sum of values of p for which the equations x+y+z=1x+2y +4z=p and x+4y +10z =`p^(2)` have a solution is `"____"`

Explanation

To solve the problem, we need to find the sum of values of \( p \) for which the equations:

1. \( x + y + z = 1 \)
2. \( x + 2y + 4z = p \)
3. \( x + 4y + 10z = p^2 \)

have a solution. We will use the determinant method to find the conditions under which these equations have solutions.

### Step 1: Set up the determinant
We will form a determinant \( \Delta \) using the coefficients of \( x, y, z \) from the equations. The determinant is given by:

\[ \Delta = \begin{vmatrix} 1 & 1 & 1 \\ 1 & 2 & 4 \\ 1 & 4 & 10 \end{vmatrix} \]