Let's start by identifying the components of the given geometric series:
[tex]\[ \frac{1}{3} + \frac{2}{9} + \frac{4}{27} + \frac{8}{81} + \frac{16}{243} \][/tex]
In a geometric series, each term is obtained by multiplying the previous term by the common ratio.
1. The first term, [tex]\( a \)[/tex], is [tex]\(\frac{1}{3}\)[/tex].
2. To find the common ratio, [tex]\( r \)[/tex], we divide the second term by the first term:
[tex]\[ r = \frac{\frac{2}{9}}{\frac{1}{3}} = \frac{2}{9} \times \frac{3}{1} = \frac{2}{3} \][/tex]
Now, we have the first term [tex]\( a = \frac{1}{3} \)[/tex] and the common ratio [tex]\( r = \frac{2}{3} \)[/tex]. We are supposed to find the sum of the first 5 terms, [tex]\( S_5 \)[/tex].
The formula for the sum [tex]\( S_n \)[/tex] of the first [tex]\( n \)[/tex] terms of a geometric series is given by:
[tex]\[ S_n = a \frac{1-r^n}{1-r} \][/tex]
For [tex]\( n = 5 \)[/tex]:
[tex]\[ S_5 = \frac{\frac{1}{3} \left(1 - \left(\frac{2}{3}\right)^5\right)}{1 - \frac{2}{3}} \][/tex]
Checking the provided options, we have:
1. [tex]\( S_5 = \frac{\frac{1}{3}\left(1-\left(\frac{2}{3}\right)^5\right)}{\left(1-\frac{2}{3}\right)} \)[/tex]
2. [tex]\( S_5 = \frac{\frac{2}{3}\left(1-\left(\frac{1}{3}\right)^5\right)}{\left(1-\frac{1}{3}\right)} \)[/tex]
From our calculations, we see that the correct equation to use is the first one because it matches our derived formula exactly.
Also, the result calculated from this equation is approximately [tex]\( 0.8683 \)[/tex].
Thus, the correct equation to use is:
[tex]\[ S_5 = \frac{\frac{1}{3}\left(1-\left(\frac{2}{3}\right)^5\right)}{\left(1-\frac{2}{3}\right)} \][/tex]
