I3.4 In the expansion of $(a x+b)^{2012}$, where $a$ and $b$ are relatively prime positive integers.
If the coefficients of $x^{\gamma}$ and $x^{\gamma+1}$ are equal, find the value of $\delta=a+b$.
Coefficient of $x^{23}=C_{23}^{2012} \cdot a^{23} \cdot b^{1989}$; coefficient of $x^{24}=C_{24}^{2012} \cdot a^{24} \cdot b^{1988}$
$$
\begin{array}{l}
C_{23}^{2012} \cdot a^{23} \cdot b^{1989}=C_{24}^{2012} \cdot a^{24} \cdot b^{1988} \\
b=\frac{C_{24}^{2012}}{C_{23}^{2012}} \cdot a \\
b=\frac{2012-24+1}{24} \cdot a \\
24 b=1989 a \\
8 b=663 a
\end{array}
$$
$\because a$ and $b$ are relatively prime integers
$$
\begin{array}{l}
\therefore a=8, b=663 \\
\delta=8+663=671
\end{array}
$$