The first term of the geometric progression is 81. The common ratio \(r\) can be calculated as \(r = \frac{54}{81} = \frac{2}{3}\).
The formula for the sum of the first \(n\) terms of a geometric progression is:
\(S_n = a \frac{1-r^n}{1-r}\)
where \(a\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms.
Substituting the given values:
\(S_{10} = 81 \frac{1-(\frac{2}{3})^{10}}{1-\frac{2}{3}}\)
\(S_{10} = 81 \frac{1-(\frac{2}{3})^{10}}{\frac{1}{3}}\)
\(S_{10} = 81 \times 3 \times (1-(\frac{2}{3})^{10})\)
\(S_{10} = 243 \times (1-(\frac{2}{3})^{10})\)
Calculating \((\frac{2}{3})^{10} \approx 0.01734\),
\(S_{10} = 243 \times (1-0.01734)\)
\(S_{10} = 243 \times 0.98266\)
\(S_{10} \approx 239\)
