jose is playing a video game in which the number of jewels that can be collected doubles every level. on level 1, there are 5 jewels that can be collected. which recursive formula gives the number of jewels jose can collect on level n?a

Answer 1

To define a recursive formula for the number of jewels that Jose can collect at level \( n \), we start by establishing the relationship between levels.

1. At level 1, the number of jewels is given as \( 5 \).
2. The number of jewels doubles at each subsequent level. Therefore:
   • At level 2, the number of jewels is \( 2 \times 5 = 10 \).
   • At level 3, the number of jewels is \( 2 \times 10 = 20 \).
   • At level 4, the number of jewels is \( 2 \times 20 = 40 \).
   • And so on.

From this, we can derive the recursive formula:

• Base case: \( J(1) = 5 \)
• Recursive case: \( J(n) = 2 \times J(n-1) \) for \( n > 1 \)

Thus, the full recursive formula is:

\[ J(n) = \begin{cases} 5 & \text{if } n = 1 \ 2 \cdot J(n-1) & \text{if } n > 1 \end{cases} \]

This formula describes how the number of jewels collected grows with each level.