Let $S(n)$ be the sum of the digits of an integer $n$. For example, $S(327)=3+2+7=12$. Find the value of

$$
A=S(1)-S(2)+S(3)-S(4)+\ldots-S(2016)+S(2017)
$$
Solution

If $m$ is even, the number $m+1$ has the same digits as $m$ except for the units digit, which is one unit larger. Therefore, $S(m+1)-S(m)=1$. This allows us to group the terms of the sequence into pairs with a difference of 1:

$$
\begin{array}{r}
S(1)-S(2)+S(3)-S(4)+\ldots-S(2016)+S(2017)= \\
S(1)+(S(3)-S(2))+(S(5)-S(4))+\ldots+(S(2017)-S(2016))= \\
1+1+1+\ldots+1= \\
1009
\end{array}
$$