Professor Célia, in a class about sequences, decides to play a guessing game with patterns:

I) First, she chooses a Natural number.

II) Cláudia must say the double of its successor.

III) Marcelo must say the triple of the predecessor of the number said by Cláudia.

IV) Finally, Ademar must say the quadruple of the successor of the number said by Marcelo.

a) If Professor Célia chooses 3, what will be the sequence formed by the 4 numbers?

b) Diani was in the bathroom and when she came back, she heard Ademar saying 184. What was the number chosen by Professor Célia?

c) Create an expression to determine the number chosen by Professor Célia if Ademar said the result is $x$.
Solution

a) If Professor Célia chooses 3, Cláudia should say $2 \cdot 4=8$, Marcelo should say 3$\cdot$7 = 21, and Ademar should say $4 \cdot 22=88$.

b) We need to analyze the problem "backwards," using inverse operations. If Ademar said 184, then Marcelo must have said $\frac{184}{4}-1=46-1=45$, and consequently, Cláudia must have said $\frac{45}{3}+1=16$. Finally, Professor Célia must have said $\frac{16}{2}-1=7$.

c) Using the reasoning from the previous item, if Ademar said $x$, then Marcelo must have said $\frac{x}{4}-1$, Cláudia must have said $\frac{\frac{x}{4}-1}{3}+1$, and the number chosen by Professor Célia was

$$
\frac{\frac{\frac{x}{4}-1}{3}+1}{2}-1=\frac{x-16}{24}
$$