Sumception 2

Let us call the quadratic equation $ax^2 - 7x + b$

We know that $\displaystyle \sum_{x=1}^{20}$ $(ax^2 - 7x + b) = 7400$

This can be evaluated to $\frac {ax(x+1)(2x+1)}{6}$ - $\frac {7x(x+1)}{2}$ + $xb$ $= 7400$

Substituting $x$ $=$ $20$ , we get the equation $2870a +20b = 8870$

We also know that $\displaystyle \sum_{x=1}^{20}$ ( $\frac {ax(x+1)(2x+1)}{6}$ - $\frac {7x(x+1)}{2}$ + $xb$ ) = $40460$

This can be evaluated to $(\frac {ax^2 (x+1)^2}{12}\ + \frac {ax(x+1)(2x+1)}{12} + \frac {ax(x+1)}{12})$ - ( $\frac {7x(x+1)(2x+1)}{12}$ + $\frac {7x(x+1)}{4}$ ) + $\frac {bx(x+1)}{2}$ $= 40460$

Substituting $x = 20$ , we get the equation $16170a+210c$ = $51240$

Solving the two equations simultaneously we can solve $a = 3$ and $b = 13$

Therefore the original equation is $3x^2 - 7x + 13$ and the third term is $\boxed{19}$