# Understanding advantage estimator in proximal policy optimization

I was reading Proximal Policy Optimization paper. It states following:

$$\hat{A}_t=-V(s_t)+r_t+\gamma r_{t+1}+...+\gamma^{T-t+1}r_{T-1}+\color{blue}{\gamma^{T-t}}V(s_T) \quad\quad\quad\quad\quad\quad\quad(10)$$
where $$t$$ specifies the time index in $$[0, T]$$, within a given length-$$T$$ trajectory segment. Generalizing this choice, we can use a truncated version of generalized advantage estimation, which reduces to Equation (10) when $$λ = 1$$:
$$\hat{A}_t=\delta_t+(\gamma\lambda)\delta_{t+1}+...+(\gamma\lambda)^{T-t+1}\delta_{T-1}\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad(11)$$
where, $$\delta_t=r_t+\gamma V(s_{t+1})-V(s_t)\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad(12)$$

How equation (11) reduces to equation (10). Putting $$\lambda=1$$ in equation (11), we get:

$$\hat{A}_t=\delta_t+\gamma\delta_{t+1}+...+\gamma^{T-t+1}\delta_{T-1}$$
Putting equation (12) in equation (11), we get:
$$\hat{A}_t$$
$$=r_t+\gamma V(s_{t+1})-V(s_t)$$
$$+\gamma[r_{t+1}+\gamma V(s_{t+2})-V(s_{t+1})]+...$$
$$+\gamma^{T-t+1}[r_{T-1}+\gamma V(s_{T})-V(s_{T-1})]$$

$$=-V(s_t)+r_t\color{red}{+\gamma V(s_{t+1})}$$
$$+\gamma r_{t+1}+\gamma^2 V(s_{t+2})\color{red}{-\gamma V(s_{t+1})}+...$$
$$+\gamma^{T-t+1}r_{T-1}+\color{blue}{\gamma^{T-t+2}} V(s_{T})-V(s_{T-1})$$

I understand the terms cancels out. I am not getting the difference in blue colored power of $$\gamma$$ in last terms. I must have made some stupid mistake.