### Instruction: What is the value of the sum $\frac{2}{3}+\frac{2^2}{3^2}+\frac{2^3}{3^3}+ \ldots +\frac{2^{10}}{3^{10}}$? Express your answer as a common fraction.### Response: I notice that this is a geometric series, where each term is obtained by multiplying the previous term by $\frac{2}{3}$.
I recall that the sum of a geometric series with first term $a$ and common ratio $r$ is given by the formula $S_n = a\frac{1-r^n}{1-r}$, where $n$ is the number of terms.
In this case, $a = \frac{2}{3}$, $r = \frac{2}{3}$, and $n = 10$.
So I can plug these values into the formula and simplify: $S_{10} = \frac{2}{3}\frac{1-(\frac{2}{3})^{10}}{1-\frac{2}{3}} = \frac{2}{3}\frac{1-(\frac{2}{3})^{10}}{\frac{1}{3}} = 2\frac{1-(\frac{2}{3})^{10}}{1}$.
To further simplify, I need to calculate $(\frac{2}{3})^{10}$.
I could use a calculator, but I also know that $(\frac{2}{3})^{10} = \frac{2^{10}}{3^{10}}$, which is the last term in the series.
So I can write $S_{10} = 2\frac{1-\frac{2^{10}}{3^{10}}}{1} = 2 - \frac{2^{11}}{3^{10}}$.
This is the exact value of the sum, but I need to express it as a common fraction.
To do that, I need to find a common denominator for the two terms.
I notice that $2 = \frac{2\cdot 3^{10}}{3^{10}}$, so I can rewrite $S_{10} = \frac{2\cdot 3^{10}}{3^{10}} - \frac{2^{11}}{3^{10}} = \frac{2\cdot 3^{10} - 2^{11}}{3^{10}}$.
This is the common fraction I was looking for.