${\left(1+x+x^2+\cdots +x^p\right)}^n=a_0+a_{1}x+a_2{x}^2+\cdots +a_{np}{x}^{np}$ $\Rightarrow a_1+2a_2+3a_3+\cdots +np.a_{np}=$

Differentiating both sides with respect to $x$

$n{\left(1+x+x^2+\cdots +x^p\right)}^{n-1}\left(1+2x+3x^2+\cdots +p{x}^{p-1}\right)=a_1+2a_{2}x+\cdots +np.a_{np}{x}^{np-1}$

$\text{put }x=1$

$n{\left(p+1\right)}^{n-1}\left(1+2+3+\cdots +p\right)=a_1+2a_2+\cdots +np{a}_{np}$

$n{\left(p+1\right)}^{n-1}.\frac{p\left(p+1\right)}{2}=a_1+2a_2+\cdots +np{a}_{np}$

$\frac{np{\left(p+1\right)}^n}{2}=a_1+2a_2+\cdots +np{a}_{np}$