Question 9
Let us define a binary operation $\star$ for positive numbers $A$ and $B$ by
$$
A \star B = \frac{A}{AB + 1}.
$$
(1) Prove that $(A \star B) \star C = A \star (B + C)$.
(2) Find the value of the expression:
$$
((((\cdots((((1 \star 2) \star 3) \star 4) \cdots) \star 59) \star 60) \star 61).
$$
$$
A \star B = \frac{A}{AB + 1}.
$$
(1) Prove that $(A \star B) \star C = A \star (B + C)$.
(2) Find the value of the expression:
$$
((((\cdots((((1 \star 2) \star 3) \star 4) \cdots) \star 59) \star 60) \star 61).
$$