Let $N_t$ a renewal process and $U(t)=E(N_t)$ Let $B_t=S_{N_t}-t$ with $S_{N_t}=X_1+\cdots+X_{N_t}$
Show $\displaystyle U(t)=\frac{t} {E(X_1)}+\frac{E(B_t)}{E(X_1)}$.
I tried: $$\frac{t} {E(X_1)}+\frac{E(B_t)}{E(X_1)}=\frac{E(S_{N_t})}{E(X_1)}=E(X_2+\cdots+X_{N_t})=E(S_{N_t-1})$$ but I don't have $U(t)$
Can you help me?
Thank you