Suppose a survey shows that a mayoral candidate is gaining votes at a rate of $2000t+1000$ votes per day, where $t$ is the number of days since she announced her candidacy.

How many supporters will the candidate have after 50 days, assuming that she had no supporters at $t=0$ ?
(Give your answer as an exact number.)
supporters:

Step 1 ：Suppose a survey shows that a mayoral candidate is gaining votes at a rate of $2000t+1000$ votes per day, where $t$ is the number of days since she announced her candidacy.

Step 2 ：We need to find out how many supporters the candidate will have after 50 days, assuming that she had no supporters at $t=0$.

Step 3 ：The number of supporters the candidate gains each day is given by the function $2000t+1000$.

Step 4 ：To find the total number of supporters after 50 days, we need to integrate this function from 0 to 50. The integral of a function gives the area under the curve, which in this case represents the total number of supporters.

Step 5 ：The integral of the function from 0 to 50 gives the total number of supporters the candidate will have after 50 days.

Step 6 ：Final Answer: The candidate will have $\boxed{{\displaystyle 2550000}}$ supporters after 50 days.