$\displaystyle Y$ have pdf $\displaystyle P(Y=k)=(k+1)p^2 q^k$, with $\displaystyle 0<p<1 $and $\displaystyle q=1-p$ and $\displaystyle k=0,1,2,...$

Is easy prove that $\displaystyle M_{Y}(t)=p^2 (1-qe^t)^2$. Let $\displaystyle X=Y_1+\cdots Y_{24}$, with $\displaystyle Y_{i}$ with same distribution of $\displaystyle Y$

Calculate $\displaystyle M_{X}(t)$

Note:$\displaystyle M_Y (t)=E(e^{tk})$