Question

Meera's character in a video game starts off a level with 122 health points. Each time an enemy hits the character, it takes health points of damage. Model the situation above with a recursive sequence. Let be the total damage Meera's character has taken before being hit for the th time.

Answer 1

To model the situation with a recursive sequence, we need to define the health points of Meera's character in relation to the damage taken from enemies.

Let's denote:

- \( H(n) \): the health points after \( n \) hits.
- \( D(n) \): the total damage taken after \( n \) hits.

Initially, Meera's character has 122 health points, which gives us the base case for our sequence:

- Base case:
\[
H(0) = 122
\]

After the first hit, Meera's health will decrease based on the damage from that hit. Let’s denote the damage taken from the \( n \)-th hit as \( d_n \). Then, we can express the health after the \( n \)-th hit recursively as:

- Recursive formula:
\[
H(n) = H(n-1) - d_n
\]

Where \( d_n \) is the damage taken during the nth hit.

To define \( D(n) \), which is the total damage taken before being hit for the \( n \)-th time, we can set it up as follows:

- Base case:
\[
D(0) = 0
\]

- Recursive formula:
\[
D(n) = D(n-1) + d_n
\]

In summary:
- The health points of Meera's character after \( n \) hits can be modeled by:
\[
H(n) = H(n-1) - d_n \quad \text{for } n \geq 1
\]
- The total damage taken before being hit for the \( n \)-th time is:
\[
D(n) = D(n-1) + d_n \quad \text{for } n \geq 1
\]

This recursive model accurately captures the mechanics of damage taken and remaining health points through successive enemy hits.