Problem (Algebra):
What is the sum of the digits of the integer equal to $\left(10^{3}+1\right)^{2}$ ?

Solution:
Using a calculator, we see that

$$
\left(10^{3}+1\right)^{2}=1001^{2}=1002001
$$

The sum of the digits of this integer is $1+2+1$ which equals 4 .

To determine this integer without using a calculator, we can let $x=10^{3}$.

Then

$$
\begin{aligned}
\left(10^{3}+1\right)^{2} & =(x+1)^{2} \\
& =x^{2}+2 x+1 \\
& =\left(10^{3}\right)^{2}+2\left(10^{3}\right)+1 \\
& =1002001
\end{aligned}
$$

Final answer: 1002001