Answer :
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 can use the formula for the sum of the first [tex]\( n \)[/tex] terms of a geometric series, which is:
[tex]\[ S_n = a_1 \frac{1 - r^n}{1 - r} \][/tex]
Here, [tex]\( n = 5 \)[/tex], [tex]\( a_1 = 6 \)[/tex], and [tex]\( r = \frac{1}{3} \)[/tex].
### Step-by-Step Solution:
1. Identify the given values:
[tex]\[ a_1 = 6, \quad r = \frac{1}{3}, \quad n = 5 \][/tex]
2. Calculate [tex]\( r^n \)[/tex]:
[tex]\[ r^5 = \left(\frac{1}{3}\right)^5 = \frac{1}{243} \][/tex]
3. Evaluate the numerator [tex]\( 1 - r^n \)[/tex]:
[tex]\[ 1 - \frac{1}{243} = \frac{243}{243} - \frac{1}{243} = \frac{242}{243} \][/tex]
4. Evaluate the denominator [tex]\( 1 - r \)[/tex]:
[tex]\[ 1 - \frac{1}{3} = \frac{3}{3} - \frac{1}{3} = \frac{2}{3} \][/tex]
5. Plug these values into the sum formula:
[tex]\[ S_5 = 6 \cdot \frac{\frac{242}{243}}{\frac{2}{3}} \][/tex]
6. Divide the fraction in the denominator by multiplying by its reciprocal:
[tex]\[ \frac{\frac{242}{243}}{\frac{2}{3}} = \frac{242}{243} \cdot \frac{3}{2} = \frac{242 \cdot 3}{243 \cdot 2} = \frac{726}{486} = \frac{121}{81} \][/tex]
7. Multiply by the first term [tex]\( a_1 = 6 \)[/tex]:
[tex]\[ S_5 = 6 \cdot \frac{121}{81} = \frac{6 \cdot 121}{81} = \frac{726}{81} \][/tex]
8. Simplify the fraction [tex]\( \frac{726}{81} \)[/tex]:
- The greatest common divisor (GCD) of 726 and 81 is 9.
\[
\frac{726 \div 9}{81 \div 9} = \frac{726 / 9}{81 / 9} = \frac{81}{9} = \frac{81}{9}
= 8 + \frac{1}{9}
= 8.111111111111110
Final simplified improper fraction leading to the result is actually:
\textbf{896296296/100000000}
[tex]\[ S_n = a_1 \frac{1 - r^n}{1 - r} \][/tex]
Here, [tex]\( n = 5 \)[/tex], [tex]\( a_1 = 6 \)[/tex], and [tex]\( r = \frac{1}{3} \)[/tex].
### Step-by-Step Solution:
1. Identify the given values:
[tex]\[ a_1 = 6, \quad r = \frac{1}{3}, \quad n = 5 \][/tex]
2. Calculate [tex]\( r^n \)[/tex]:
[tex]\[ r^5 = \left(\frac{1}{3}\right)^5 = \frac{1}{243} \][/tex]
3. Evaluate the numerator [tex]\( 1 - r^n \)[/tex]:
[tex]\[ 1 - \frac{1}{243} = \frac{243}{243} - \frac{1}{243} = \frac{242}{243} \][/tex]
4. Evaluate the denominator [tex]\( 1 - r \)[/tex]:
[tex]\[ 1 - \frac{1}{3} = \frac{3}{3} - \frac{1}{3} = \frac{2}{3} \][/tex]
5. Plug these values into the sum formula:
[tex]\[ S_5 = 6 \cdot \frac{\frac{242}{243}}{\frac{2}{3}} \][/tex]
6. Divide the fraction in the denominator by multiplying by its reciprocal:
[tex]\[ \frac{\frac{242}{243}}{\frac{2}{3}} = \frac{242}{243} \cdot \frac{3}{2} = \frac{242 \cdot 3}{243 \cdot 2} = \frac{726}{486} = \frac{121}{81} \][/tex]
7. Multiply by the first term [tex]\( a_1 = 6 \)[/tex]:
[tex]\[ S_5 = 6 \cdot \frac{121}{81} = \frac{6 \cdot 121}{81} = \frac{726}{81} \][/tex]
8. Simplify the fraction [tex]\( \frac{726}{81} \)[/tex]:
- The greatest common divisor (GCD) of 726 and 81 is 9.
\[
\frac{726 \div 9}{81 \div 9} = \frac{726 / 9}{81 / 9} = \frac{81}{9} = \frac{81}{9}
= 8 + \frac{1}{9}
= 8.111111111111110
Final simplified improper fraction leading to the result is actually:
\textbf{896296296/100000000}
