2024 AMC 8 Problems/Problem 8

Problem

On Monday Taye has $2. Every day, he either gains $3 or doubles the amount of money he had on the previous day. How many different dollar amounts could Taye have on Thursday, 3 days later?

$\textbf{(A) } 3\qquad\textbf{(B) } 4\qquad\textbf{(C) } 5\qquad\textbf{(D) } 6\qquad\textbf{(E) } 7$

==Solution 1== (BRUTE FORCE) How many values could be on the first day? Only $2$ dollars. The second day, you can either add $3$ dollars, or double, so you can have $5$ dollars, or $4$ . For each of these values, you have $2$ values for each. For $5$ dollars, you have $10$ dollars or $8$ , and for $4$ dollars, you have $8$ dollars or $7$ . Now, you have $2$ values for each of these. For $10$ dollars, you have $13$ dollars or $20$ , for $8$ dollars, you have $16$ dollars or $11, for$ 8 $dollars, you have$ 16 $dollars or$ 11 $, and for$ 7 $dollars, you have$ 14 $dollars or$ 10 $.$ {$11}$ (Error compiling LaTeX. Unknown error_msg),${$ (Error compiling LaTeX. Unknown error_msg)16} $repeat leaving you with$ 8-2 = \boxed{(C) 6}$ different values