G4.1 Let $b=1^{2}-2^{2}+3^{2}-4^{2}+5^{2}-\ldots-2012^{2}+2013^{2}$. Determine the remainder of $b$ divided by 2015 .
$$
\begin{array}{l}
b=1+(3-2)(3+2)+(5-4)(5+4)+\ldots+(2013-2012)(2012+2013) \\
b=1+5+9+\ldots+4025
\end{array}
$$

This an arithmetic series, $a=1, d=4,1+(n-1) \times 4=4025 \Rightarrow n=1007$
$$
b=\frac{1007}{2}(1+4025)=1007 \times 2013=1007 \times(2015-2)=1007 \times 2015-2014=1006 \times 2015+1
$$

Remainder $=1$