The cumulative reward = rt+1 (rt+k+1 = rt+0+1 = rt+1)+ rt+2 (rt+k+1 = rt+1+1 = rt+2) + ...

It seems like you're asking about the formula for cumulative reward in the context of reinforcement learning. The cumulative reward is the total reward a learning agent receives over a period of time. The formula you've provided seems to be a bit mixed up, but the general idea is correct.

The cumulative reward, often denoted as G_t, at time step t is calculated as the sum of all future rewards the agent will receive, each discounted by a factor of gamma raised to the power of the time step at which the reward is received.

Here's the correct formula:

G_t = r_(t+1) + gamma * r_(t+2) + gamma^2 * r_(t+3) + ...

Where:

• G_t is the cumulative reward at time step t
• r_(t+1), r_(t+2), r_(t+3), ... are the rewards received at each future time step
• gamma is the discount factor, which determines the present value of future rewards.

This formula assumes that the agent's goal is to maximize the sum of the discounted future rewards. The discount factor gamma is between 0 and 1 - if it's close to 0, the agent will prioritize immediate rewards, while if it's close to 1, the agent will prioritize long-term rewards.