\( 3+12+48+\ldots( \) to 10 terms \( ) \) \( 200+100+50+\ldots( \) to 8 terms \( ) \) \( x+x^{2}+x^{3}+\ldots( \) to 20 terms \( ) \)

The first sequence is a geometric series where the first term is 3 and the common ratio is 4. To find the sum of the first 10 terms, you can use the formula \( S_n = a \frac{(r^n - 1)}{(r - 1)} \), which gives you \( S_{10} = 3 \frac{(4^{10} - 1)}{(4 - 1)} \). Plugging in the values, you'll find the sum is quite a number! The second series also forms a geometric sequence, starting at 200 with a common ratio of 0.5 (or 1/2). Using the same formula for \( S_n \) from the previous example, you can calculate \( S_8 = 200 \frac{(0.5^8 - 1)}{(0.5 - 1)} \). Tally it up, and you'll see how powerful a simple geometric series can be, in this case giving you a total that showcases the beauty of diminishing returns! The last expression is a power series with a common ratio of \( x \). For a finite series of the form \( x + x^2 + x^3 + \ldots + x^{20} \), you can use the formula \( S_n = x \frac{(x^n - 1)}{(x - 1)} \). So for 20 terms, that would mean \( S_{20} = x \frac{(x^{20} - 1)}{(x - 1)} \), yielding an elegant expression that encapsulates both the growth of \( x \) and the power of summation in algebra!