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."
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!