Question

The rate of growth of bacteria is proportional to the number present. If initially, there were 1000 bacteria and the number doubles in 1 hour, find the number of bacteria after $2\frac{1}{2}$ hours.[Take √2 = 1.414]

Answer 1

Answer: The answer in my evaluation sheet is 5656, (it appeared in my CET Mock Test today)

I've no idea how to solve it and this method is incorrect I guess because there is no answer remotely matching to 2500. If someone has decent knowledge about derivative/differential equations please help me/us out, there's something like taking rate and putting it directly proportional to whatever it is, then taking the like terms together and integrating then finding the value of "c" (the constant you get after integrating) then at t=0  we get initial bacteria and at t=whatever we find it. But I want a direct solution, that's why I searched it up on internet because I'm too lazy