Q:

Most Expeditious Way of Adding Consecutive Composite Numbers

In helping my 10-year old son with a homework problem that he was trying to solve by rote, I found myself resorting to arithmetic series. In short, I was calculating the arithmetic series from 12 to 37 and then subtracting the primes. He has not done series yet in school.
My questions is two-part: 
1) Was I really using the fastest, most elegant way to solve it?
2) Is there a way to teach him without using series?
Here was the actual question: "Find the sum of all composite numbers between 12 and 37."

A:

How i would go about this will be:
$$
\begin{array}{rl}
&S&=  12 + 13 +...+ 36 + 37\\
+&S&=  37 + 36 +...+ 13 + 12\\
\hline
&2S&= 49 + 49 +...+ 49 + 49\\
&&= 49\cdot (37-12+1)\\
&&= 49\cdot 26
\end{array}
$$
$$
\begin{align}
S&= 49 \cdot 13\\
&= (50-1)\cdot13\\
 &= 650 -13\\
& = 637\\
\end{align}
$$
Now, the primes between 12 and 37 are $13, 17, 23, 27,29,31, 37$ that have a sum of 177.
Thus, required $\text{sum}= 637-177= 460$.
A slight alteration could be to exclude 37 from the sum from the start itself. Hope this helps!

