Calculate the few first terms of the sequence that described the health points of the monster, and find a recursive rule for the sequence.
It will take 260 seconds to kill the monster.
Practice makes perfect
Let a_n be the number of health points (HP) of the final monster in the game after the n^(th) time that 10 seconds pass. The monster starts with 100 HP. This tells us that a_0=100. The monster regains 10 % of its HP after every 10 seconds, and it loses 10 points during this 10 seconds. Let's find a recursive rule for a_n.
a_n - Health of Monster
At the beginning of the n^(th) round the monster has a_(n-1) HP. During this round it loses 10 points, therefore he has a_(n-1)-10 HP. At the end of the round it regains 10 % of its HP. This tells us that we should multiply a_(n-1)-10 by 110 %=1.1 to get a_n.
a_0=100
a_n=1.1(a_(n-1)-10)
⇕
a_n=1.1a_(n-1)-11
Now, let's use a spreadsheet to find when the monster is defeated. Let's enter a_0=100 in cell A1 and write =Round((1.1)*(A1)-11,2) in cell A2. When we hit enter we will get the following.
Next, copy cell A2, highlight cells A3 through A50, and paste. Notice that the sequence drops below 0 around cell A27. The monster's HP after the first round is in cell A2. This tells us that HP after the n^(th) round is in the (n+1)^(th) row.
In cell A26 there is the number 1.69. This tells us that a_(25)=1.69. Therefore, after 25* 10=250 seconds the monster has only 1.69 HP. In the next 10 seconds Helena deals 10 damage and kills the monster. Finally, the monster is defeated after 250+10=260 seconds.
