My name is Tamara. The
level of my question is middle, and I am the student. The master needs some of the trees (twenty, to be exact) at the back of his spooky old mansion cleared to make way for a new evil laboratory, so he decides to send some his slaves to do the work for him. He initially sends out four of his men, armed with axes, to chop the trees down. Due to the fact he is very impatient, every ten minutes he sends out another man to help with the work. Assuming that it takes one man 30 minutes to chop down 1/3 of a tree, how long till all twenty trees are chopped down? 

Hi Tamara, Since the new men go out every 10 minites, I am going to work in 10 minute time intervals. Hence I am going to read your last statement as "Assuming that it takes one man 10 minutes to chop down 1/9 of a tree, how long till all twenty trees are chopped down?" In the first 10 minutes there are 4 men chopping and hence 4/9 of a tree gets cut down.In the next 10 minutes there are 4+1 men working and hence 4/9+1/9 of a tree is chopped down. In the next 10 minutes there are 4+2 men working and hence 4/9+2/9 of a tree is chopped down..... Hence you can see that the number of trees chopped down is
I suppose that you could keep adding terms, do the arithmetic at each setp, and count how many 10 minute time intervals it takes until the sum is 20. But this is an arithmetic series, and if you know something about arithmetic series you can find an easier way.
You may know a formula for the sum of an arithmetic series that you can use here, but I don't usually remember formulas. What I remember is that you write down the series twice, once forward and once backward and then add "down".
Adding "down" I get
which is
Multiplying both sides by 9 and simplifying this becomes
This doesn't factor but if you use the general quadratic you can find the number of 10 minute time intervals it takes to chop down 20 trees. Cheers, 

