To find the sum of the first five terms of a geometric series with the first term [tex]\( a_1 = 6 \)[/tex] and common ratio [tex]\( r = \frac{1}{3} \)[/tex], we use the formula for the sum of the first [tex]\( n \)[/tex] terms of a geometric series:
[tex]\[ S_n = a_1 \frac{1 - r^n}{1 - r} \][/tex]
Let's break down the steps:
1. The first term ([tex]\( a_1 \)[/tex]) of the series is [tex]\( 6 \)[/tex].
2. The common ratio ([tex]\( r \)[/tex]) is [tex]\( \frac{1}{3} \)[/tex].
3. The number of terms ([tex]\( n \)[/tex]) is [tex]\( 5 \)[/tex].
Substitute these values into the formula:
[tex]\[ S_5 = 6 \frac{1 - (\frac{1}{3})^5}{1 - \frac{1}{3}} \][/tex]
Calculate [tex]\( (\frac{1}{3})^5 \)[/tex]:
[tex]\[ (\frac{1}{3})^5 = \frac{1}{243} \][/tex]
Now the formula looks like this:
[tex]\[ S_5 = 6 \frac{1 - \frac{1}{243}}{1 - \frac{1}{3}} \][/tex]
Calculate the denominator [tex]\( 1 - \frac{1}{3} \)[/tex]:
[tex]\[ 1 - \frac{1}{3} = \frac{2}{3} \][/tex]
Now substitute back into the formula:
[tex]\[ S_5 = 6 \frac{1 - \frac{1}{243}}{\frac{2}{3}} \][/tex]
Calculate [tex]\( 1 - \frac{1}{243} \)[/tex]:
[tex]\[ 1 - \frac{1}{243} = \frac{242}{243} \][/tex]
The formula further simplifies to:
[tex]\[ S_5 = 6 \frac{\frac{242}{243}}{\frac{2}{3}} \][/tex]
To divide by a fraction, multiply by its reciprocal:
[tex]\[ S_5 = 6 \times \frac{242}{243} \times \frac{3}{2} \][/tex]
Simplify the multiplication:
[tex]\[ S_5 = 6 \times \frac{242 \times 3}{243 \times 2} \][/tex]
Multiply the numerators and the denominators:
[tex]\[ S_5 = 6 \times \frac{726}{486} \][/tex]
Now, simplify the fraction:
[tex]\[ \frac{726}{486} \div 6 = \frac{121}{81} \][/tex]
Thus:
[tex]\[ S_5 = \frac{242}{27} \][/tex]
So, the sum of the first five terms of the geometric series, expressed as an improper fraction in the lowest terms, is:
[tex]\[ \boxed{\frac{242}{27}} \][/tex]
