This solution comes from Henry. Well done!

To make a square, we need three more matchsticks.

To make a second square, we need three more matchsticks.

I noticed that for each square we make we need three matchsticks, in addition to the one we had at the beginning. Therefore to make 10 squares we need a total of 31 matchsticks. For 20 squares we need 61 matchsticks and for 50 squares we need 151.

The general pattern isif $n$ is the number of squares, we use $3n + 1$ matchsticks in total.