Question 7
The Binomial Expansion is valid for exponents that are not integers. That is, for all real numbers $x,$ $y,$ and $r$ with $|x| \gt |y|,$ $$
\begin{aligned} &(x + y)^r \\ &= x^r + r x^{r-1} y \\ &{}+ \frac{r(r - 1)}{2!}\,x^{r-2} y^2 \\ &{}+ \frac{r(r - 1)(r - 2)}{3!}\,x^{r-3} y^3 \\ &{}+ \cdots \end{aligned}
$$ What are the first three digits to the right of the decimal point in the decimal representation of $\left(10^{2002} + 1\right)^{\frac{10}{7}}?$
\begin{aligned} &(x + y)^r \\ &= x^r + r x^{r-1} y \\ &{}+ \frac{r(r - 1)}{2!}\,x^{r-2} y^2 \\ &{}+ \frac{r(r - 1)(r - 2)}{3!}\,x^{r-3} y^3 \\ &{}+ \cdots \end{aligned}
$$ What are the first three digits to the right of the decimal point in the decimal representation of $\left(10^{2002} + 1\right)^{\frac{10}{7}}?$