No Calculators allowed.

The left-hand side of the equation is equivalent to

$\sum_{k=1}^{2n}$ ( $k^{3}$ ) - ${8}$ $\sum_{j=1}^{n}$ ( $k^{3}$ ) =

$[(2n) * (2n + 1) / 2]^{2}\ - 8 * [n * (n + 1) / 2]^{2}$ =

$n^{2} * (2n^{2} - 1)$ .

Setting this equal to ${1413721}$ and solving for $n^{2}$ using the quadratic equation gives us $n^{2} = {841}$ , and thus $n = \boxed{29}$ .

THIS SOLUTION IS ONLY HELPFUL TO THOSE WHO LOVE TO CODE. BUT I'D BE GLAD IF SOMEONE COULD ALSO PROVIDE A MATHEMATICAL APPROACH TO THIS PROBLEM.

//include iostream statement

//include cmath statement

using namespace std;

int series (int pos) {

int result = (pos * 2);

result -= 1;

result = pow (result, 3);

return (result);

}

int main (int argc, char* argv []) {

int result (0);

int counter (0);

while (result != 1413721) {

counter++;

result += series (counter);

}

cout << counter << endl;

return (0);

}

COMPILE WITH ANY C++ COMPILER TO GET THE ANSWER, i.e. 29

NOT A SOLUTION. Note: The idea behind this problem came from me, then me and @Daniel Liu worked on it (ok it was pretty easy) and i found the numbers.

nice solution Brian.