Let $$a, b, c$$   $$ \in $$ $$R$$ . If $$f\left( x \right) = a{x^2} + bx + c$$     is such that $$a + b + c = 3{\text{ and }}f\left( {x + y} \right) = f\left( x \right) + f\left( y \right) + xy,\,\forall \,x,y \in R,{\text{ then }}\sum\limits_{n = 1}^{10} {f\left( n \right)} {\text{ is equal to:}}$$

Solution :
$$\eqalign{ & f\left( x \right) = a{x^2} + bx + c \cr & f\left( 1 \right) = a + b + c = 3 \cr & \Rightarrow f\left( 1 \right) = 3 \cr & {\text{Now }}f\left( {x + y} \right) = f\left( x \right) + f\left( y \right) + xy\,\,\,\,\,\,.....\left( 1 \right) \cr & {\text{Put }}x = y = 1{\text{ in eqn}}{\text{.}}\left( 1 \right) \cr & f\left( 2 \right) = f\left( 1 \right) + f\left( 1 \right) + 1 \cr & \,\,\,\,\,\,\,\,\,\,\,\, = 2f\left( 1 \right) + 1 \cr & f\left( 2 \right) = 7 \cr & \Rightarrow \,f\left( 3 \right) = 12 \cr & {\text{Now, }}{S_n} = 3 + 7 + 12 + ...... + {t_{n\,\,\,\,}}\,\,\,\,\,\,.....\left( 1 \right) \cr & {S_n} = 3 + 7 + ...... + {t_{n - 1}} + {t_n}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,.....\left( 2 \right) \cr & {\text{Subtract}}\left( 2 \right){\text{from}}\left( 1 \right) \cr & {t_n} = 3 + 4 + 5 + ......{\text{ upto }}n{\text{ terms}} \cr & {t_n} = \frac{{\left( {{n^2} + 5n} \right)}}{2} \cr & {S_n} = \sum {{t_n}\, = \sum {\frac{{\left( {{n^2} + 5n} \right)}}{2}} } \cr & {S_n} = \frac{1}{2}\left[ {\frac{{n\left( {n + 1} \right)\left( {2n + 1} \right)}}{6} + \frac{{5n\left( {n + 1} \right)}}{2}} \right] \cr & \,\,\,\,\,\, = \frac{{n\left( {n + 1} \right)\left( {n + 8} \right)}}{6} \cr & {S_{10}} = \frac{{10 \times 11 \times 18}}{6} = 330 \cr} $$